Question Find the $y$ -intercept of linear function $y = \frac{3}{4}x + 2$ and compare it to the $y$ -intercepts of the points for function B. Determine which statement is true about the $y$ -intercepts.

Studdy Solution

STEP 1

Assumptions
1. Function A is given by the equation $y = \frac{3}{4}x + 2$ .
2. Function B is defined by a table of values for $x$ and $y$ .
3. We need to find the $y$ -intercept and the rate of change (slope) for both functions.
4. The $y$ -intercept is the value of $y$ when $x = 0$ .
5. The rate of change (slope) for a linear function is the ratio of the change in $y$ to the change in $x$ (rise over run).

STEP 2

Identify the $y$ -intercept of Function A directly from its equation.
The $y$ -intercept of a function in slope-intercept form $y = mx + b$ is $b$ .

STEP 3

For Function A, the $y$ -intercept $b$ is given as 2.

STEP 4

To find the $y$ -intercept of Function B, we need to determine the value of $y$ when $x = 0$ .

STEP 5

Examine the table for Function B to see if there is a value for $y$ when $x = 0$ . If not, we will calculate the slope and use it to find the $y$ -intercept.

STEP 6

Since there is no entry for $x = 0$ in the table for Function B, we will calculate the slope using two points from the table.

STEP 7

Choose two points from the table of Function B to calculate the slope. Let's use $(-3, 2)$ and $(-1, -2)$ .

STEP 8

Calculate the slope for Function B using the formula for slope $m = \frac{\Delta y}{\Delta x}$ .

STEP 9

Compute the change in $y$ ( $\Delta y$ ) for Function B using the chosen points.
$\Delta y = y_2 - y_1 = -2 - 2$

STEP 10

Compute the change in $x$ ( $\Delta x$ ) for Function B using the chosen points.
$\Delta x = x_2 - x_1 = -1 - (-3)$

STEP 11

Now calculate the slope for Function B.
$m_B = \frac{\Delta y}{\Delta x} = \frac{-2 - 2}{-1 - (-3)}$

STEP 12

Simplify the calculation for the slope of Function B.
$m_B = \frac{-4}{2} = -2$

STEP 13

Now that we have the slope of Function B, we can use one of the points and the slope to find the $y$ -intercept.

STEP 14

Use the point-slope form of a line to find the $y$ -intercept of Function B. The point-slope form is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is a point on the line.

STEP 15

Using the point $(-3, 2)$ and the slope $-2$ , write the equation for Function B.
$y - 2 = -2(x - (-3))$

STEP 16

Simplify the equation to find the $y$ -intercept of Function B.
$y - 2 = -2(x + 3)$

STEP 17

Expand the right side of the equation.
$y - 2 = -2x - 6$

STEP 18

Add 2 to both sides of the equation to solve for $y$ .
$y = -2x - 4$

STEP 19

Now we can identify the $y$ -intercept of Function B as $-4$ .

STEP 20

Compare the $y$ -intercepts of Function A and Function B.
Function A has a $y$ -intercept of 2, and Function B has a $y$ -intercept of $-4$ .

STEP 21

Determine which statement is true based on the comparison of $y$ -intercepts.
The $y$ -intercept of Function A (2) is greater than the $y$ -intercept of Function B ( $-4$ ).

STEP 22

Now let's compare the rates of change (slopes) of Function A and Function B.

STEP 23

The slope of Function A is $\frac{3}{4}$ as given in the equation.

STEP 24

The slope of Function B is $-2$ as calculated from the table.

STEP 25

Compare the slopes of Function A and Function B.
Function A has a slope of $\frac{3}{4}$ and Function B has a slope of $-2$ .

STEP 26

Determine which statement is true based on the comparison of slopes.
The rate of change for Function A is not equal to the rate of change for Function B.

STEP 27

Based on the comparisons made in steps 20 and 25, the correct statement is:
C. The $y$ -intercept of Function A is greater than the $y$ -intercept of Function B.
The solution is option C.

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Question Find the yyy-intercept of linear function y=34x+2y = \frac{3}{4}x + 2y=43​x+2 and compare it to the yyy-intercepts of the points for function B. Determine which statement is true about the yyy-intercepts.

Studdy Solution

STEP 1

STEP 2

Identify the yyy-intercept of Function A directly from its equation.The yyy-intercept of a function in slope-intercept form y=mx+b y = mx + b y=mx+b is b b b.

STEP 3

For Function A, the yyy-intercept b b b is given as 2.

STEP 4

To find the yyy-intercept of Function B, we need to determine the value of y y y when x=0 x = 0 x=0.

STEP 5

Examine the table for Function B to see if there is a value for y y y when x=0 x = 0 x=0. If not, we will calculate the slope and use it to find the yyy-intercept.

STEP 6

Since there is no entry for x=0 x = 0 x=0 in the table for Function B, we will calculate the slope using two points from the table.

STEP 7

Choose two points from the table of Function B to calculate the slope. Let's use (−3,2) (-3, 2) (−3,2) and (−1,−2) (-1, -2) (−1,−2).

STEP 8

Calculate the slope for Function B using the formula for slope m=ΔyΔx m = \frac{\Delta y}{\Delta x} m=ΔxΔy​.

STEP 9

Compute the change in y y y (Δy \Delta y Δy) for Function B using the chosen points.Δy=y2−y1=−2−2 \Delta y = y_2 - y_1 = -2 - 2 Δy=y2​−y1​=−2−2

STEP 10

Compute the change in x x x (Δx \Delta x Δx) for Function B using the chosen points.Δx=x2−x1=−1−(−3) \Delta x = x_2 - x_1 = -1 - (-3) Δx=x2​−x1​=−1−(−3)

STEP 11

Now calculate the slope for Function B.mB=ΔyΔx=−2−2−1−(−3) m_B = \frac{\Delta y}{\Delta x} = \frac{-2 - 2}{-1 - (-3)} mB​=ΔxΔy​=−1−(−3)−2−2​

STEP 12

Simplify the calculation for the slope of Function B.mB=−42=−2 m_B = \frac{-4}{2} = -2 mB​=2−4​=−2

STEP 13

Now that we have the slope of Function B, we can use one of the points and the slope to find the yyy-intercept.

STEP 14

Use the point-slope form of a line to find the yyy-intercept of Function B. The point-slope form is y−y1=m(x−x1) y - y_1 = m(x - x_1) y−y1​=m(x−x1​), where m m m is the slope and (x1,y1) (x_1, y_1) (x1​,y1​) is a point on the line.

STEP 15

Using the point (−3,2) (-3, 2) (−3,2) and the slope −2 -2 −2, write the equation for Function B.y−2=−2(x−(−3)) y - 2 = -2(x - (-3)) y−2=−2(x−(−3))

STEP 16

Simplify the equation to find the yyy-intercept of Function B.y−2=−2(x+3) y - 2 = -2(x + 3) y−2=−2(x+3)

STEP 17

Expand the right side of the equation.y−2=−2x−6 y - 2 = -2x - 6 y−2=−2x−6

STEP 18

Add 2 to both sides of the equation to solve for y y y.y=−2x−4 y = -2x - 4 y=−2x−4

STEP 19

Now we can identify the yyy-intercept of Function B as −4 -4 −4.

STEP 20

Compare the yyy-intercepts of Function A and Function B.Function A has a yyy-intercept of 2, and Function B has a yyy-intercept of −4 -4 −4.

STEP 21

Determine which statement is true based on the comparison of yyy-intercepts.The yyy-intercept of Function A (2) is greater than the yyy-intercept of Function B (−4-4−4).

STEP 22

Now let's compare the rates of change (slopes) of Function A and Function B.

STEP 23

The slope of Function A is 34 \frac{3}{4} 43​ as given in the equation.

STEP 24

The slope of Function B is −2 -2 −2 as calculated from the table.

STEP 25

Compare the slopes of Function A and Function B.Function A has a slope of 34 \frac{3}{4} 43​ and Function B has a slope of −2 -2 −2.

STEP 26

Determine which statement is true based on the comparison of slopes.The rate of change for Function A is not equal to the rate of change for Function B.

STEP 27

Based on the comparisons made in steps 20 and 25, the correct statement is:C. The yyy-intercept of Function A is greater than the yyy-intercept of Function B.The solution is option C.

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Question Find the $y$ -intercept of linear function $y = \frac{3}{4}x + 2$ and compare it to the $y$ -intercepts of the points for function B. Determine which statement is true about the $y$ -intercepts.

Identify the $y$ -intercept of Function A directly from its equation.
The $y$ -intercept of a function in slope-intercept form $y = mx + b$ is $b$ .

For Function A, the $y$ -intercept $b$ is given as 2.

To find the $y$ -intercept of Function B, we need to determine the value of $y$ when $x = 0$ .

Examine the table for Function B to see if there is a value for $y$ when $x = 0$ . If not, we will calculate the slope and use it to find the $y$ -intercept.

Since there is no entry for $x = 0$ in the table for Function B, we will calculate the slope using two points from the table.

Choose two points from the table of Function B to calculate the slope. Let's use $(-3, 2)$ and $(-1, -2)$ .

Calculate the slope for Function B using the formula for slope $m = \frac{\Delta y}{\Delta x}$ .

Compute the change in $y$ ( $\Delta y$ ) for Function B using the chosen points.
$\Delta y = y_2 - y_1 = -2 - 2$

Compute the change in $x$ ( $\Delta x$ ) for Function B using the chosen points.
$\Delta x = x_2 - x_1 = -1 - (-3)$

Now calculate the slope for Function B.
$m_B = \frac{\Delta y}{\Delta x} = \frac{-2 - 2}{-1 - (-3)}$

Simplify the calculation for the slope of Function B.
$m_B = \frac{-4}{2} = -2$

Now that we have the slope of Function B, we can use one of the points and the slope to find the $y$ -intercept.

Use the point-slope form of a line to find the $y$ -intercept of Function B. The point-slope form is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is a point on the line.

Using the point $(-3, 2)$ and the slope $-2$ , write the equation for Function B.
$y - 2 = -2(x - (-3))$

Simplify the equation to find the $y$ -intercept of Function B.
$y - 2 = -2(x + 3)$

Expand the right side of the equation.
$y - 2 = -2x - 6$

Add 2 to both sides of the equation to solve for $y$ .
$y = -2x - 4$

Now we can identify the $y$ -intercept of Function B as $-4$ .

Compare the $y$ -intercepts of Function A and Function B.
Function A has a $y$ -intercept of 2, and Function B has a $y$ -intercept of $-4$ .

Determine which statement is true based on the comparison of $y$ -intercepts.
The $y$ -intercept of Function A (2) is greater than the $y$ -intercept of Function B ( $-4$ ).

The slope of Function A is $\frac{3}{4}$ as given in the equation.

The slope of Function B is $-2$ as calculated from the table.

Compare the slopes of Function A and Function B.
Function A has a slope of $\frac{3}{4}$ and Function B has a slope of $-2$ .

Determine which statement is true based on the comparison of slopes.
The rate of change for Function A is not equal to the rate of change for Function B.

Based on the comparisons made in steps 20 and 25, the correct statement is:
C. The $y$ -intercept of Function A is greater than the $y$ -intercept of Function B.
The solution is option C.