Question

function a and function b are linear functions.
function a

x	y
-9	-14
-3	-4
9	16

function b (graph of a line on a coordinate plane)
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 equal to 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:

Step1: Calculate slope of Function A

Use slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Take points \((-9, -14)\) and \((-3, -4)\):
\( m_A = \frac{-4 - (-14)}{-3 - (-9)} = \frac{10}{6} = \frac{5}{3} \).

Step2: Calculate slope of Function B

From graph, two points: \((0, 3)\) (y-intercept) and \((3, 8)\) (approx).
\( m_B = \frac{8 - 3}{3 - 0} = \frac{5}{3} \). So \( m_A = m_B \), first statement true.

Step3: Find equation of Function A

Using point-slope: \( y - (-14) = \frac{5}{3}(x - (-9)) \) → \( y + 14 = \frac{5}{3}(x + 9) \) → \( y = \frac{5}{3}x + 15 - 14 \) → \( y = \frac{5}{3}x + 1 \).
At \( x = -6 \): \( y_A = \frac{5}{3}(-6) + 1 = -10 + 1 = -9 \).

Step4: Find equation of Function B

From graph, y-intercept \( b = 3 \), slope \( \frac{5}{3} \), so \( y = \frac{5}{3}x + 3 \).
At \( x = -6 \): \( y_B = \frac{5}{3}(-6) + 3 = -10 + 3 = -7 \).

Step5: Analyze statements

- Slopes: \( m_A = m_B = \frac{5}{3} \) → first statement true, second false.
- At \( x = -6 \): \( y_A = -9 \), \( y_B = -7 \). So \( y_A < y_B \) → third (equal) and fourth (A > B) false. Wait, correction: Wait, earlier calculation for Function A's equation: Wait, recalculate Function A's equation. Using point \((-9, -14)\):
\( y = \frac{5}{3}x + c \). Plug \( x = -9 \), \( y = -14 \):
\( -14 = \frac{5}{3}(-9) + c \) → \( -14 = -15 + c \) → \( c = 1 \). Correct. At \( x = -6 \): \( y = \frac{5}{3}(-6) + 1 = -10 + 1 = -9 \).
Function B: From graph, when \( x = 0 \), \( y = 3 \); when \( x = -3 \), \( y = \frac{5}{3}(-3) + 3 = -5 + 3 = -2 \)? Wait, no, earlier point was wrong. Let's take two clear points from graph: ( -2, 0 ) and ( 1, 5 ). Then \( m = \frac{5 - 0}{1 - (-2)} = \frac{5}{3} \), correct. Y-intercept: when \( x = 0 \), \( y = \frac{5}{3}(0) + b \), from ( -2, 0 ): \( 0 = \frac{5}{3}(-2) + b \) → \( b = \frac{10}{3} ≈ 3.33 \). Wait, my initial y-intercept was wrong. Let's use two points from graph: ( -2, 0 ) and ( 1, 5 ). So equation \( y = \frac{5}{3}x + \frac{10}{3} \). At \( x = -6 \): \( y = \frac{5}{3}(-6) + \frac{10}{3} = \frac{-30 + 10}{3} = \frac{-20}{3} ≈ -6.67 \). Wait, this is conflicting. Wait, original graph: Let's check the graph again. The line passes through ( -2, 0 ) and ( 4, 10 ). So slope \( \frac{10 - 0}{4 - (-2)} = \frac{10}{6} = \frac{5}{3} \). Y-intercept: when \( x = 0 \), \( y = \frac{5}{3}(0) + b \). From ( -2, 0 ): \( 0 = \frac{5}{3}(-2) + b \) → \( b = \frac{10}{3} ≈ 3.33 \). So at \( x = -6 \): \( y = \frac{5}{3}(-6) + \frac{10}{3} = \frac{-30 + 10}{3} = \frac{-20}{3} ≈ -6.67 \). But Function A at \( x = -6 \): \( y = \frac{5}{3}(-6) + 1 = -10 + 1 = -9 \). So \( y_A = -9 \), \( y_B ≈ -6.67 \). So \( y_A < y_B \). So the correct true statements:
1. The slope of Function A is equal to the slope of Function B (since both \( \frac{5}{3} \)).
The other statements:
- "The slope of Function A is greater than..." → false.
- "y-value of A at x=-6 equal to B" → false.
- "y-value of A at x=-6 greater than B" → false. Wait, but initial calculation had error in Function B's y-intercept. Let's re-express:

Correct steps:
Function A:
Points: (-9, -14), (-3, -4), (9, 16). Slope between (-9, -14) and (-3, -4): \( \frac{-4 - (-14)}{-3 - (-9)} = \frac{10}{6} = \frac{5}{3} \). Correct. Equation: \( y = \frac{5}{3}x + 1 \) (as before).

Function B:
From graph, let's take two clear points: ( -2, 0 ) and ( 4, 10 ). Slope: \( \frac{10 - 0}{4 - (-2)} = \frac{10}{6} = \frac{5}{3} \). Correct. Equation: Using point ( -2, 0 ): \( y - 0 = \frac{5}{3}(x + 2) \) → \( y = \frac{5}{3}x + \frac{10}{3} \).

At \( x = -6 \):
Function A: \( y = \frac{5}{3}(-6) + 1 = -10 + 1 = -9 \).
Function B: \( y = \frac{5}{3}(-6) + \frac{10}{3} = -10 + \frac{10}{3} = \frac{-30 + 10}{3} = \frac{-20}{3} ≈ -6.666... \).

So:
- Slopes: equal (both \( \frac{5}{3} \)) → first statement true.
- Slope of A > slope of B? No, equal → second false.
- y-value of A at x=-6 equal to B? -9 vs -6.666: No → third false.
- y-value of A at x=-6 greater than B? -9 < -6.666: No → fourth false. Wait, but the original checkmarks in the image are wrong. Wait, maybe I misread the graph. Let's check the graph again. The Function B line: when x=0, y=3? Wait, looking at the graph, the line passes through (0, 3) and (3, 8)? Wait, 0 to 3 on x, y from 3 to 8: 8-3=5, 3-0=3, slope 5/3. Then at x=-6: y = (5/3)(-6) + 3 = -10 + 3 = -7. Ah! Here's the mistake: earlier, I took the wrong y-intercept. The graph's y-intercept is 3, not 10/3. Let's re-express Function B:

From the graph, the line crosses the y-axis at (0, 3) (since at x=0, y=3). Then another point: (3, 8) (x=3, y=8). Slope: (8-3)/(3-0)=5/3. Correct. So equation: \( y = \frac{5}{3}x + 3 \).

Now, at x=-6:
Function B: \( y = \frac{5}{3}(-6) + 3 = -10 + 3 = -7 \).

Function A: \( y = \frac{5}{3}x + 1 \). At x=-6: \( y = \frac{5}{3}(-6) + 1 = -10 + 1 = -9 \).

So:
- Slopes: both 5/3 → first statement true.
- Slope of A > slope of B? No, equal → second false.
- y-value of A at x=-6 equal to B? -9 vs -7: No → third false.
- y-value of A at x=-6 greater than B? -9 < -7: No → fourth false.

But the original image has checkmarks on all, which is incorrect. Wait, maybe the Function A's equation was miscalculated. Let's recalculate Function A's slope with another pair: (-3, -4) and (9, 16). Slope: (16 - (-4))/(9 - (-3)) = 20/12 = 5/3. Correct. Equation: using (-3, -4): \( y - (-4) = \frac{5}{3}(x - (-3)) \) → \( y + 4 = \frac{5}{3}(x + 3) \) → \( y = \frac{5}{3}x + 5 - 4 \) → \( y = \frac{5}{3}x + 1 \). Correct.

Function B: y-intercept 3, slope 5/3, equation \( y = \frac{5}{3}x + 3 \). Correct.

At x=-6:
A: -9, B: -7. So A's y is less than B's.

So the only true statement is "The slope of Function A is equal to the slope of Function B."

Answer:

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