Question

function a and function b are linear functions.
function a
y = \frac{1}{2}x + 1
function b
graph of a line on a coordinate plane with x-axis from -10 to 10 and y-axis from -10 to 10, passing through points, y-intercept at 4
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 greater than the slope of function b.

Explanation:

Step1: Find y-intercept of Function A

The equation of Function A is \( y = \frac{1}{2}x + 1 \). For a linear function in the form \( y = mx + b \), the y-intercept is \( b \). So, the y-intercept of Function A is \( 1 \).

Step2: Find y-intercept of Function B

From the graph of Function B, when \( x = 0 \), \( y = 4 \). So, the y-intercept of Function B is \( 4 \).

Step3: Compare y-intercepts

Since \( 1 < 4 \), the y-intercept of Function A is less than the y-intercept of Function B.

Step4: Find slope of Function A

The equation of Function A is \( y = \frac{1}{2}x + 1 \), so the slope \( m_A = \frac{1}{2} \).

Step5: Find slope of Function B

Using two points on Function B's graph, e.g., \( (0, 4) \) and \( (2, 6) \). The slope \( m_B = \frac{6 - 4}{2 - 0} = \frac{2}{2} = 1 \).

Step6: Compare slopes

Since \( \frac{1}{2} < 1 \), the slope of Function A is less than the slope of Function B.

Answer:

The true statements are:

• 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.