Question

function a and function b are linear functions.
function a
graph of a line in a coordinate plane with x from -10 to 10 and y from -10 to 10, passing through points, with a purple line
function b

x	y
-9	-2
6	8
9	10

select all the statements that are true.

• the slope of function a is equal to the slope of function b.
• the slope of function a is greater than the slope of function b.
• the y - value of function a when x = -6 is less than the y - value of function b when x = -6.
• the y - value of function a when x = -6 is greater than the y - value of function b when x = -6.

Explanation:

To solve this, we analyze the slopes and \( y \)-values of the linear functions.

Step 1: Find the slope of Function A

Function A is a line. From the graph, we identify two points:

• When \( x = 0 \), \( y = -5 \) (y-intercept).
• When \( x = 8 \), \( y = -1 \).

The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
Substitute \( (x_1, y_1) = (0, -5) \) and \( (x_2, y_2) = (8, -1) \):

$$ m_A = \frac{-1 - (-5)}{8 - 0} = \frac{4}{8} = \frac{1}{2} $$

Step 2: Find the slope of Function B

Function B has the table:

\( x \)	\( y \)
\( 6 \)	\( 8 \)

Use two points, e.g., \( (-9, -2) \) and \( (6, 8) \).
Slope formula:

$$ m_B = \frac{8 - (-2)}{6 - (-9)} = \frac{10}{15} = \frac{2}{3} $$

Step 3: Compare the slopes

\( m_A = \frac{1}{2} = 0.5 \) and \( m_B = \frac{2}{3} \approx 0.666 \).
Thus, \( m_A < m_B \).

Step 4: Find \( y \)-value of Function A at \( x = -6 \)

Function A is linear, so we use its equation. From Step 1, the y-intercept is \( -5 \), so the equation is \( y = \frac{1}{2}x - 5 \).
Substitute \( x = -6 \):

$$ y_A = \frac{1}{2}(-6) - 5 = -3 - 5 = -8 $$

Step 5: Find \( y \)-value of Function B at \( x = -6 \)

Function B is linear. First, find its equation. Using \( m_B = \frac{2}{3} \) and point \( (-9, -2) \):

$$ y - (-2) = \frac{2}{3}(x - (-9)) \implies y + 2 = \frac{2}{3}(x + 9) $$

Simplify:

$$ y = \frac{2}{3}x + 6 - 2 \implies y = \frac{2}{3}x + 4 $$

Substitute \( x = -6 \):

$$ y_B = \frac{2}{3}(-6) + 4 = -4 + 4 = 0 $$

Step 6: Compare \( y \)-values at \( x = -6 \)

\( y_A = -8 \) and \( y_B = 0 \). Thus, \( y_A < y_B \).

Evaluate the Statements

1. "The slope of Function A is equal to the slope of Function B."

\( m_A = \frac{1}{2} \), \( m_B = \frac{2}{3} \) (not equal). False.

1. "The slope of Function A is greater than the slope of Function B."

\( m_A = 0.5 < m_B \approx 0.666 \). False.

1. "The \( y \)-value of Function A when \( x = -6 \) is less than the \( y \)-value of Function B when \( x = -6 \)."

\( y_A = -8 \), \( y_B = 0 \) ( \( -8 < 0 \) ). True.

1. "The \( y \)-value of Function A when \( x = -6 \) is greater than the \( y \)-value of Function B when \( x = -6 \)."

\( -8 < 0 \) (not greater). False.

Answer:

The only true statement is:

• The \( y \)-value of Function A when \( x = -6 \) is less than the \( y \)-value of Function B when \( x = -6 \).