Question

function a and function b are linear functions.
function a
graph of function a: a line on a coordinate grid with x from -10 to 10 and y from -10 to 10, passing through points, y-intercept at (0, 2)
function b
$y = \frac{5}{2}x - 1$
select all the statements that are true.

• the y-intercept of function a is less than the y-intercept of function b.
• the slope of function a is less than the slope of function b.
• the y-intercept of function a is greater than the y-intercept of function b.
• the slope of function a is equal to the slope of function b.

Explanation:

Step1: Find y-intercept of Function A

From the graph, Function A crosses the y-axis at (0, 2), so its y-intercept is 2.

Step2: Find y-intercept of Function B

Function B is \( y = \frac{5}{2}x - 1 \), so its y-intercept is -1.

Step3: Compare y-intercepts

Since \( 2 > -1 \), the y-intercept of Function A is greater than that of Function B.

Step4: Find slope of Function A

Using two points on Function A, e.g., (0, 2) and (-2, 0). Slope \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 2}{-2 - 0} = \frac{-2}{-2} = 1 \).

Step5: Find slope of Function B

Function B has slope \( \frac{5}{2} = 2.5 \).

Step6: Compare slopes

Since \( 1 < 2.5 \), the slope of Function A is less than that of Function B. Wait, no—wait, earlier y-intercept comparison: Function A's y-intercept is 2, Function B's is -1, so 2 > -1 (so "The y-intercept of Function A is greater than the y-intercept of Function B" is true). Slope of A is 1, slope of B is 2.5, so "The slope of Function A is less than the slope of Function B" is true? Wait, no, let's recheck. Wait, the options:

First option: "The y-intercept of Function A is less than the y-intercept of Function B" — 2 < -1? No, false.

Second option: "The slope of Function A is less than the slope of Function B" — 1 < 2.5? Yes, true.

Third option: "The y-intercept of Function A is greater than the y-intercept of Function B" — 2 > -1? Yes, true.

Fourth option: "The slope of Function A is equal to the slope of Function B" — 1 = 2.5? No, false.

Wait, but the initial check: Let's re-express Function A's equation. From the graph, when x=0, y=2 (y-intercept 2). Let's take another point, like (2, 6) (since from (0,2), moving 2 right, 4 up? Wait, no, earlier point (-2, 0): (0,2) to (-2,0): change in y is -2, change in x is -2, so slope 1. Then (2, 4)? Wait, no, maybe I made a mistake. Wait, the graph: when x=2, what's y? Let's see the grid. The line goes through (0,2) and (2, 6)? Wait, no, if slope is 2? Wait, maybe my initial point selection was wrong. Let's take (0,2) and (1, 4)? Wait, no, the graph: from (0,2), moving 1 right, 2 up? Wait, no, the line passes through (-1, 0)? Wait, no, the x-intercept is at x=-1? Wait, no, the graph shows the line crossing the x-axis at x=-1? Wait, no, looking at the graph: the line goes through (0, 2) and (-2, -2)? Wait, no, maybe I misread the graph. Let's re-express Function A. Let's use two clear points. Let's take (0, 2) and (2, 6). Then slope is (6 - 2)/(2 - 0) = 4/2 = 2. Oh! I made a mistake earlier. So (0,2) and (2,6): slope is 2. Then Function B's slope is 5/2 = 2.5. So slope of A is 2, slope of B is 2.5. So slope of A (2) is less than slope of B (2.5). Y-intercept of A is 2, y-intercept of B is -1. So:

- "The y-intercept of Function A is less than the y-intercept of Function B": 2 < -1? No, false.

- "The slope of Function A is less than the slope of Function B": 2 < 2.5? Yes, true.

- "The y-intercept of Function A is greater than the y-intercept of Function B": 2 > -1? Yes, true.

- "The slope of Function A is equal to the slope of Function B": 2 = 2.5? No, false.

Wait, so two true statements: "The slope of Function A is less than the slope of Function B" and "The y-intercept of Function A is greater than the y-intercept of Function B". But let's check the options again. The user's options:

1. The y-intercept of Function A is less than the y-intercept of Function B. (False)

2. The slope of Function A is less than the slope of Function B. (True)

3. The y-intercept of Function A is greater than the y-intercept of Function B. (True)

4. The slope of Function A is equal to the slope of Function B. (False)

But maybe the initial graph reading was wrong. Let's re-express Function A's equation. Let's use the y-intercept (0, 2) and another point. Let's take (1, 4): slope is (4-2)/(1-0)=2. So equation is y = 2x + 2. Function B is y = (5/2)x - 1. So slope of A is 2, slope of B is 2.5. Y-intercept of A is 2, B is -1. So:

- Y-intercept A (2) > Y-intercept B (-1): true (option 3).

- Slope A (2) < Slope B (2.5): true (option 2).

But the user's initial check had some boxes checked. Wait, the problem is to select all true statements. So the true ones are:

- The slope of Function A is less than the slope of Function B.

- The y-intercept of Function A is greater than the y-intercept of Function B.

Wait, but let's confirm again.

Function A: from graph, when x=0, y=2 (y-intercept 2). Let's take x=2: y=6 (since 2*2 + 2=6). So slope is (6-2)/(2-0)=2.

Function B: y = (5/2)x - 1, slope 5/2=2.5, y-intercept -1.

So:

- Y-intercept A (2) vs B (-1): 2 > -1 → option 3 is true.

- Slope A (2) vs B (2.5): 2 < 2.5 → option 2 is true.

So the true statements are:

- The slope of Function A is less than the slope of Function B.

- The y-intercept of Function A is greater than the y-intercept of Function B.

Answer:

The true statements are:

• The slope of Function A is less than the slope of Function B.
• The y - intercept of Function A is greater than the y - intercept of Function B.

(In boxed form for the options, if we consider the options as:

1. The slope of Function A is less than the slope of Function B.
2. The y - intercept of Function A is greater than the y - intercept of Function B.

So the answer is the options with these statements.)