2x + y = 2
x - y = 1

[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$850\\ r=rate\to 2.6\%\to \frac{2.6}{100}\dotfill &0.026\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\dotfill &4\\ t=years\dotfill &5 \end{cases} \\\\\\ A = 850\left(1+\frac{0.026}{4}\right)^{4\cdot 5}\implies A=850(1.0065)^{20} \implies A \approx 967.60[/tex]

Answer:

Solution: x = 2, y = -3.

Step-by-step explanation:

Given system of equations:

Equation 1: 2x - 3y = 13

Equation 2: x + 2y = -4

Multiply Equation 2 by 2:

2(x + 2y) = 2(-4)

Simplified: 2x + 4y = -8

Add Equation 1 and Equation 3:

4x + y = 5

Solve Equation 4 for y:

y = 5 - 4x

Substitute y into Equation 2:

x + 2(5 - 4x) = -4

Simplify:

-7x + 10 = -4

Solve for x:

x = 2

Substitute x into Equation 5:

y = 5 - 4(2)

y = -3

Solution: x = 2, y = -3.

Mean (Juniors) = Sum / number of data points = 1085 / 12 ≈ 90.42

Median (Juniors) = (95 + 95) / 2 = 95

Mean (Seniors) = Sum / number of data points = 1095 / 12 ≈ 91.25

Median (Seniors) = (80 + 80) / 2 = 80

Juniors: 125, 90, 95, 80, 100, 100, 120, 85, 95, 120, 95, 85

First, sort the data: 80, 85, 85, 90, 95, 95, 95, 100, 100, 120, 120, 125

Sum = 1085

Mean (Juniors) = Sum / number of data points = 1085 / 12 ≈ 90.42

Median (Juniors) = (95 + 95) / 2 = 95

Seniors: 135, 70, 65, 100, 35, 175, 80, 80, 90, 65, 120, 60

First, sort the data: 35, 65, 65, 70, 80, 80, 90, 100, 120, 120, 135, 175

Sum = 1095

Mean (Seniors) = Sum / number of data points = 1095 / 12 ≈ 91.25

Median (Seniors) = (80 + 80) / 2 = 80

Variability

Juniors:

Range = max - min = 125 - 80 = 45

Seniors:

Range = max - min = 175 - 35 = 140

C. Based on the mean and median, it appears that seniors spend slightly more time on homework on average (mean ≈ 91.25) compared to juniors (mean ≈ 90.42).

However, the median time spent for seniors (80 minutes) is less than that of juniors (95 minutes), indicating that the data for seniors might be skewed by some outliers.

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The numeric value of the expression 3y(9 - x) when y = 3 and x = 7 is given as follows:

18.

To obtain the numeric value of a function or even of an expression, we must substitute each instance of the variable of interest on the function by the value at which we want to find the numeric value of the function or of the expression presented in the context of a problem.

The expression in the context of this problem is given as follows:

3y(9 - x).

Hence the numeric value when x = 7 and y = 3 is given as follows:

3 x 3 x (9 - 7) = 3 x 3 x 2 = 18.

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The ratio t : u : v in its simplest form is 0.4 : 1 : v/0.4.

To find the ratio t : u : v, we'll need to express each variable in terms of one another. Given the information provided, we know that:

T = 0.4u   (T is 40% of u)

Y = 0.5t   (Y is half of t)

We can substitute the value of t from the second equation into the first equation:

T = 0.4u

Y = 0.5(0.4u) = 0.2u

Now, we have the values of T and Y in terms of u. To find v, we need additional information since it is not provided.

Without that information, we cannot determine the specific value of v or its relationship to the other variables. Therefore, we cannot simplify the ratio t : u : v any further.

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Answer:

-3/4, answer D

Step-by-step explanation:

Slope = rise/run. The equation is (y2-y1)/(x2-x1).

So let's pick 2 points. There are pink dots on (-2,2) and (6,-4), so we'll use those points.

slope = (2-(-4))/(-2-6)

slope = (2+4)/(-2-6)

slope = (6)/-8, which we can simplify by dividing the numerator and denominator each by 2.

slope = -3/4

Just as a quick check, this makes sense because as x increases, y decreases so we should expect the slope to be negative.

Answer:

Step-by-step explanation:

Use 2 points from the line:  (-2, 2) and (6, -4)

slope = y2 -y1  =   -4 - 2  = -6   = - 3

             x2 - x1       6 - -2      8         4

The slope is choice  D    -3/4

The value of x will be 10.

Given a triangle we need to set a proportion to solve for the value of x,

Therefore, according to the definition of the similar triangles,

x / 5 = 15+5 / x

x² = 5 × 20

x² = 100

x = 10

Hence the value of x will be 10.

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Answer:

5. 94 square units.

6. 159 square units.

Step-by-step explanation:

By using Heron's Formula: [tex]\bold{\sqrt{s(s-a)(s-b)(s-c)}}[/tex]

5.

Using the values c = 17, b = 13, and a = 15,

Calculate the semi-perimeter of the triangle as:

s =[tex]\frac{a+b+c}{2}[/tex]

s =[tex]\frac{15 + 13 + 17}{2}=22.5[/tex]

Using Heron's formula, the area of the triangle is:

A = [tex]\bold{\sqrt{s(s-a)(s-b)(s-c)}}[/tex]

A = [tex]\sqrt{22.5(22.5-15)(22.5-13)(22.5-17)}=93.90[/tex] ≈94 approximately

Therefore, the area of triangle ABC is approximately 94 square units.

6.

Using the values, a = 23, b = 25, and c = 14,

we can calculate the semi-perimeter of the triangle as:

s =[tex]\frac{a+b+c}{2}[/tex]

s =[tex]\frac{ 23 + 25 + 14}{2}[/tex]

s = 31

Using Heron's formula, the area of the triangle is:

A = [tex]\bold{\sqrt{s(s-a)(s-b)(s-c)}}[/tex]

A = [tex]\sqrt{31(31-23)(31-25)(31-14)}=159.05[/tex]≈159 approximately

Therefore, the area of triangle ABC is approximately 159 square units.

Answer:

5)  93.90 square units (2 d.p.)

6)  159.05 square units (2 d.p.)

Step-by-step explanation:

To find the area of a triangle given its side lengths, use Heron's formula.

[tex]\boxed{\begin{minipage}{8 cm}\underline{Heron's Formula}\\\\$A=\sqrt{s(s-a)(s-b)(s-c)}$\\\\where:\\ \phantom{ww}$\bullet$ $A$ is the area of the triangle. \\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the side lengths of the triangle. \\ \phantom{ww}$\bullet$ $s$ is half the perimeter.\\\end{minipage}}[/tex]

[tex]\hrulefill[/tex]

Given side lengths of triangle ABC:

To find the half perimeter, s, half the sum of the 3 side lengths:

[tex]\implies s=\dfrac{a+b+c}{2}=\dfrac{15+13+17}{2}=22.5[/tex]

Substitute the values of a, b, c and s into Heron's formula and solve for area, A:

[tex]\begin{aligned}A&=\sqrt{s(s-a)(s-b)(s-c)}\\&=\sqrt{22.5(22.5-15)(22.5-13)(22.5-17)}\\&=\sqrt{22.5(7.5)(9.5)(5.5)}\\&=\sqrt{8817.1875}\\&=93.89988019...\\&=93.90\; \sf units^2\;(2\;d.p.)\end{aligned}[/tex]

Therefore, the area of triangle ABC is 93.90 square units, to two decimal places.

[tex]\hrulefill[/tex]

Given side lengths of triangle ABC:

To find the half perimeter, s, half the sum of the 3 side lengths:

[tex]\implies s=\dfrac{a+b+c}{2}=\dfrac{23+25+14}{2}=31[/tex]

Substitute the values of a, b, c and s into Heron's formula and solve for area, A:

[tex]\begin{aligned}A&=\sqrt{s(s-a)(s-b)(s-c)}\\&=\sqrt{31(31-23)(31-25)(31-14)}\\&=\sqrt{31(8)(6)(17)}\\&=\sqrt{25296}\\&=159.0471628...\\&=159.05\; \sf units^2\;(2\;d.p.)\end{aligned}[/tex]

Therefore, the area of triangle ABC is 159.05 square units, to two decimal places.

The graph that has the same end behavior as f(x)= (10x³+x²+3)/(2x²+1) is Option A. See the attached.

the end behavior of the graph of f(x) is:

As x → +∞ , f(x) → +∞ (approaches positive infinity).

As x → -∞,  f(x) → -∞   (approaches negative infinity).

End  behavior of a function f  defines the behavior of the function's graph at the "ends" of the x- axis.

In other words, the end behavior of a function explains the graph's trend when  we look at the right end of the x-axis (as x approaches +∞) and the left end of the x -axis (as x approaches - ∞ ).

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Answer:

[tex]\dfrac{-1}{2}[/tex]

Step-by-step explanation:

          (-2 , 3) ⇒ x₁ = -2 & y₁ = 3

          (0 , 2) ⇒  x₂ = 0 & y₂ = 2

[tex]\boxed{\bf Slope = \dfrac{y_2-y_1}{x_2-x_1}}[/tex]

            [tex]\sf = \dfrac{2-3}{0-[-2]}\\\\\\=\dfrac{-1}{0+2}\\\\=\dfrac{-1}{2}[/tex]

The value of the given angle which is N and that is rounded to the nearest hundredth would be =45.58°

To calculate the missing angle N, the sine formula should be used and it is given below;

= a/sin A = b/sin B

Where;

a= 7

A= 90°

b= 5

B= N

That is;

7/sin90°= 5/sinN

sinN= 5×1/7

= 5/7

= 0.714285714

N = Sin-1(0.714285714)

= 45.58°

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Answer:

Store A

Step-by-step explanation:

If an item is 20% off, then you pay 80% of the price.

If an item is 10% off, then you pay 90% of the price.

Store A

80% of $1640 = 0.8 × $1640 = $1312

Store B

90% of $1500 = 0.9 × $1500 = $1350

Answer: Store A

Answer:

Store A

Step-by-step explanation:

Store A offers a 20% discount which is equivalent to 0.20 x $1,640 = $328.

The laptop is therefore sold for $1,640 - $328 = $1,312 at Store A.

Store B offers a 10% discount which is equivalent to 0.10 x $1,500 = $150.

The laptop is therefore sold for $1,500 - $150 = $1,350 at Store B.

Therefore, Store A offers a better deal at $1,312 compared to Store B at $1,350.

The probability that a random adult has a high school degree or some college, but has no college degree is 53. 50 % .

The number of adults with some high school degree or some college would be :

= 3, 561 + 6, 058

= 9, 619 people

The probability that a random adult would have high school degree or some college is therefore :

= Adults with high school degree or some college / Total adults in the town

= 9, 619 / 17, 981

= 53. 50 %

In conclusion, the probability is 53. 50%.

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Answer:

maybe a

Step-by-step explanation:

as t percent of 60 is 3

so it will show 57 or 63

The function is a quadratic function, and none of the given options is a factor of the graph of the function.

Here,, we have,

to determine the factor:

The zeros of the function are given as:

-3 and 5

This means that:

x = -3 and x = 5

Rewrite as:

x + 3 = 0 and x - 5 = 0

Multiply both equations

(x + 3)(x -5) = 0 * 0

This gives

(x + 3)(x -5) = 0

Express as a function

f(x) = (x + 3)(x -5)

The above means that the factors of the function are (x + 3) and (x -5)

Hence, none of the given options is a factor of the graph of the function

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complete question:

Graph of a function, f(x), has zeros at - 3, and 5. Which of

following is a factor of f(x)?

(2x - 3)

(2x + 3)

(3x + 2)

(x + 5)

Answer:

 3.98

Step-by-step explanation:

Answer:

the second method may be through graph

The graph of the proportional relationship is a linear graph, with slope of 4.6 and intercept of zero.

A proportional relationship is a relationship in which a constant ratio between the output variable and the input variable is present.

The equation that defines the proportional relationship is a linear function with slope k and intercept zero given as follows:

y = kx.

The slope k is the constant of proportionality, representing the increase or decrease in the output variable y when the constant variable x is increased by one.

Hence the graph is a linear function with slope of k and intercept of zero.

The cost of 4 pints is of $18.4, and the cost of 10 pints is of $46, hence the slope of the relation is given as follows:

k = 46/10 = 18.4/4 = 4.6.

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Jorge's gross earnings for the week assembling 317 game boards would be $633.

To calculate Jorge's gross earnings for the week, we need to determine how many game boards fall into each pay rate category and then multiply the quantity by the corresponding pay rate. Let's break down the calculation step by step:

Determine the number of game boards that fall into each pay rate category:

First 100 boards: 100 boards (as the rate applies to the first 100 boards).

Next 200 boards: Since Jorge assembled 317 boards in total and the first 100 boards are already accounted for, we subtract 100 from 317 to get 217 boards that fall into this category.

Over 300 boards: We check if the total number of boards exceeds 300. In this case, since Jorge assembled 317 boards, we have

317 - 300 = 17

boards that fall into this category.

Calculate the earnings for each category by multiplying the number of boards by the corresponding pay rate:

Earnings for the first 100 boards:

100 boards * $1.82 per board = $182.

Earnings for the next 200 boards:

217 boards * $1.92 per board = $416.64.

Earnings for the over 300 boards:

17 boards * $2.08 per board = $35.36.

Sum up the earnings from each category to find the gross pay for the week:

Gross pay = Earnings for the first 100 boards + Earnings for the next 200 boards + Earnings for the over 300 boards

Gross pay = $182 + $416.64 + $35.36

Gross pay = $633.

Therefore, Jorge's gross earnings for the week assembling 317 game boards would be $633.

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The probability that the next chew Fatoumata removes from the bag will be orange or apple, rounded to the nearest whole number, is approximately 55%.

To calculate the probability that the next chew Fatoumata removes from the bag will be either orange or apple, we need to determine the total number of times an orange or apple chew was selected.

From the given results, Fatoumata selected an orange chew 6 times and an apple chew 10 times. To find the probability, we add the number of occurrences for each chew:

Orange chews: 6 times

Apple chews: 10 times

Total occurrences of orange or apple chews: 6 + 10 = 16

Fatoumata performed the experiment a total of 29 times. So, the probability of selecting an orange or apple chew can be calculated as:

Probability = (Total occurrences of orange or apple chews) / (Total number of experiments) = 16 / 29

To express this probability as a percentage, we multiply the probability by 100:

Probability (as a decimal) = 16 / 29 ≈ 0.5517

Probability (as a percentage) = 0.5517 × 100 ≈ 55.17%

Therefore, the probability that the next chew Fatoumata removes from the bag will be orange or apple, rounded to the nearest whole number, is approximately 55%.

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1. The marginal product of the second worker is 150 units.

2. The total output of five workers is 75 units.

3. The marginal product of the sixth worker is 405 units.

4. The average total cost of producing 425 units of output is $30.

5. The average variable cost of producing 400 units of output is $20.

6. The average fixed cost of producing 430 units of output is $0.07.

The marginal product of the second worker:

= (Output of 2 workers - Output of 1 worker)

= (250 - 100)

= 150 units

The total output of five workers is:

= Output of 4 workers + Marginal product of the fifth worker

= 50 + 25

= 75 units

The marginal product of the sixth worker is:

= (Output of 6 workers - Output of 5 workers)

= (430 - 25)

= 405 units.

The average total cost to produce 425 units is:

= (Total variable cost + Total fixed cost) / Total output

The total variable cost is $30 * 425 = $12,750

The total fixed cost is $5 * 6 = $30

Total cost = $12,750 + $30 = $12,780

The Average total cost is:

$12,780 / 425

= $30

The average variable cost to produce 400 units is:

= Total variable cost / total output

Total variable cost is $20 * 400 = $8,000

Average variable cost = $8,000 / 400 = $20

The average fixed cost to produce 430 units is:

= Total fixed cost / Total output

Total fixed cost is $5 * 6 = $30

The average fixed cost is:

= $30 / 430

= $0.07.

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The angle between A and B is 90 degrees.

The angle between two vectors A and B, you can use the dot product formula:

A · B = [tex]|A| |B| cos(\theta)[/tex]

where A · B is the dot product of A and B, |A| is the magnitude of A, |B| is the magnitude of B, and theta is the angle between the two vectors.

First, let's calculate the dot product of A and B:

A · B = (3)(4) + (2)(-3) + (-6)(1)

= 12 - 6 - 6

= 0

Next, let's calculate the magnitudes of vectors A and B:

|A| = √((3)² + (2)² + (-6)²)

= √(9 + 4 + 36)

= √(49)

= 7

|B| = √((4)² + (-3)² + (1)²)

= √(16 + 9 + 1)

= √(26)

Now, we can substitute these values into the formula to find the angle theta:

0 = [tex](7)(\sqrt(26)) cos(\theta)[/tex]

Since the dot product is 0, we know that the angle between A and B is either 90 degrees or 270 degrees.

To determine which one it is, we can look at the sign of the dot product. If it's positive, the angle is acute (less than 90 degrees) and if it's negative, the angle is obtuse (greater than 90 degrees).

The dot product is 0, so the angle is orthogonal (perpendicular) and is exactly 90 degrees.

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The given quadrilateral is a parallelogram by the theorem "The diagonals of a parallelogram bisect each other."

Given is a quadrilateral.

We have to determine that the quadrilateral is a parallelogram or not.

From the shape and the drawing given, the quadrilateral is a parallelogram.

A parallelogram has opposite sides parallel to each other.

Here it is parallel.

Here it is given that the diagonals are bisected by each other, that is two parts of the diagonals are of equal length.

We have a theorem that "The diagonals of a parallelogram bisect each other."

A square, rectangle, rhombus can be included in the group of parallelogram.

So this is a parallelogram.

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The correct matching of each compound to its type:

8.1 CO - oxide

8.2 CO₂ - dioxide

8.3 SO₂ - dioxide

8.4 CaCO - incorrect notation (needs a subscript for the number of atoms)

8.5 NaOH - hydroxide

8.1 CO - oxide

8.2 CO₂ - dioxide

8.3 SO₂ - dioxide

8.4 CaCO - carbonate

8.5 NaOH - hydroxide

Here is the correct matching of each compound to its type:

8.1 CO - oxide

8.2 CO₂ - dioxide

8.3 SO₂ - dioxide

8.4 CaCO - incorrect notation (needs a subscript for the number of atoms)

8.5 NaOH - hydroxide

Compound 8.4, CaCO, is not correctly represented. It should have a subscript to indicate the number of atoms present in the compound.

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Answer:

  240 in²

Step-by-step explanation:

You want the lateral area of a triangular prism with triangular base having sides 6 in, 8 in, and 10 in, and a distance between them of 10 in.

The lateral area of a triangular prism is the area of the three rectangular faces. The width of those faces is the "height" of the prism, given as 10 in. The total length of those faces is the perimeter of the base:

  6 in + 8 in + 10 in = 24 in

The lateral area is ...

  A = LW = (24 in)(10 in) = 240 in²

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The specific data points or the Frequency distribution of grades for the 40 students.

The specific information from the graph , the exact number of students who received a grade of 9.

If the graph represents the distribution of grades of 40 students on a scale from 7 to 10, we can estimate the number of students who received a grade of 9 by considering the shape of the graph.

the graph is a normal distribution or bell curve, with the majority of students clustered around the mean (which is likely to be around 8.5 or 9), we can expect a significant number of students to have received a grade of 9. In a typical bell curve distribution, the highest density of scores is around the mean.

without additional information about the shape of the graph or the specific values represented on the x-axis, it is difficult to provide an accurate estimation of the number of students who received a grade of 9. The graph may represent a variety of distributions, such as skewed, bimodal, or uniform.

the exact number of students who received a grade of 9,  the specific data points or the frequency distribution of grades for the 40 students.

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Based on the information given regard to the graph, there are 6 students who got 9.

A graph is a diagram such as a series of one or more points, lines, line segments, curves, or areas which represents the variation of a variable in comparison with that of one or more other variables.

The collection of all points whose coordinates satisfy a given relation such as a function.

The graph shows that there are 4 students who got 7, 8 students who got 8, and 12 students who got 10. This leaves:

40 - 4 - 8 - 12

= 6 students who got 9.

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Answer:12,890 centimeters

Step-by-step explanation: 1 meter equals 100 centimeters.

The required, 128.9 meters is equal to 12,890 centimeters, as following the unit system.

To convert meters to centimeters, we can use the conversion factor that states there are 100 centimeters in one meter.

Therefore, to find out how many centimeters are equal to 128.9 meters, we multiply the number of meters by 100:

128.9 meters * 100 centimeters/meter = 12,890 centimeters

In summary, when converting meters to centimeters, we multiply the number of meters by 100 to obtain the equivalent length in centimeters. In this case, 128.9 meters is equal to 12,890 centimeters.

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The factored of [tex]7a^{10} - 112a^2[/tex] is: [tex]7a^2(a^8 - 16)[/tex]

To factor the expression [tex]7a^{10} - 112a^2[/tex]  we can identify the greatest common factor (GCF) and factor it out. Let's break down the steps:

The coefficients 7 and 112 have a common factor of 7.

The lowest power of "a" in each term.

In this case, the lowest power of "a" is 2.

Factoring out [tex]7a^2[/tex] from both terms, we get:

[tex]7a^2(a^8 - 16)[/tex]

Therefore, the factored form of [tex]7a^{10} - 112a^2[/tex] is: [tex]7a^2(a^8 - 16)[/tex]

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The cost of packaging per weight unit is given by (1/4) (x / (x + 2)) x.

We have,

To find the cost of packaging per weight unit, we need to calculate the cost of packaging (C) and divide it by the weight of the chocolates (W).

Given:

Cost of packaging (C) = 1/4 x²

Weight of chocolates (W) = x + 2

To find the cost of packaging per weight unit, we can express it as C/W.

Substituting the given expressions:

C/W = (1/4 x²) / (x + 2)

To simplify the expression, we can first simplify the numerator:

C/W = (1/4 x²) / (x + 2)

= (1/4) x (x² / (x + 2))

= (1/4) x (x(x) / (x + 2))

= (1/4) x (x / (x + 2)) * x

Now, we can rewrite the expression as:

C/W = (1/4) x (x / (x + 2)) x

Therefore,

The cost of packaging per weight unit is given by (1/4) (x / (x + 2)) x.

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The solution for X is X ≈ [tex]tan^{(-1)(0.7429)[/tex], which is approximately 0.6619 radians or approximately 37.89 degrees.

Given: 1 secX + 3 = cosX + tanX (2 + sinX)

To simplify the equation, let's express secX and tanX in terms of sinX and cosX:

1/cosX + 3 = cosX + sinX/cosX (2 + sinX)

Now, multiply both sides by cosX to eliminate the denominators:

[tex]1 + 3cosX = cos^2X + sinX(2 + sinX)[/tex]

Expanding the equation:

[tex]1 + 3cosX = cos^2X + 2sinX + sin^2X[/tex]

Since [tex]sin^2X + cos^2X[/tex] equals 1 (based on the Pythagorean identity), we can substitute it into the equation:

[tex]1 + 3cosX = 1 + 2sinX + sin^2X[/tex]

Rearranging the terms:

[tex]3cosX = 2sinX + sin^2X[/tex]

Now, let's express cosX in terms of sinX using the Pythagorean identity:

cosX = √(1 - [tex]sin^2X[/tex])

Substituting this into the equation:

3√(1 - [tex]sin^2X) = 2sinX + sin^2X[/tex]

Squaring both sides to eliminate the square root:

[tex]9(1 - sin^2X) = (2sinX + sin^2X)^2[/tex]

Expanding the right side:

[tex]9(1 - sin^2X) = (2sinX)^2 + 2(2sinX)(sin^2X) + (sin^2X)^2\\9 - 9sin^2X = 4sin^2X + 4sin^3X + sin^4X[/tex]

Rearranging the terms:

[tex]sin^4X + 4sin^3X + 4sin^2X + 9sin^2X - 9 = 0[/tex]

Combining like terms:

[tex]sin^4X + 4sin^3X + 13sin^2X - 9 = 0[/tex]

Unfortunately, this equation does not have a simple algebraic solution. It needs to be solved numerically or graphically. Using numerical methods or a graphing calculator, we can find that one solution to this equation is sinX ≈ 0.7429.

Taking the inverse tangent (arctan) of both sides, we have:

X = arctan(0.7429)

Therefore, the solution for X is X ≈ t[tex]an^{(-1)}(0.7429)[/tex], which is approximately 0.6619 radians or approximately 37.89 degrees.

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The function g(x) in the form a(x-h)^2 + k is: [tex]g(x) = (x + 2)^2 - 4[/tex]

Starting with[tex]f(x) = x^2[/tex], the translation 2 units left and 4 units down would result in the following transformation:

g(x) = f(x + 2) - 4

Substituting[tex]f(x) = x^2:[/tex]

[tex]g(x) = (x + 2)^2 - 4[/tex]

Expanding the square:

[tex]g(x) = x^2 + 4x + 4 - 4[/tex]

Simplifying:

[tex]g(x) = x^2 + 4x[/tex]

Now we need to rewrite this expression in the form [tex]a(x-h)^2 + k.[/tex] To do this, we will complete the square:

[tex]g(x) = x^2 + 4x\\g(x) = (x^2 + 4x + 4) - 4\\g(x) = (x + 2)^2 - 4[/tex]

Therefore, the function g(x) in the form a(x-h)^2 + k is:

[tex]g(x) = (x + 2)^2 - 4[/tex]

Where a = 1, h = -2, and k = -4.

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