Question

function a and function b are linear functions.
function a
x | y
-10 | -2
-5 | -1
5 | 1
function b
(graph of a line on a coordinate plane with x from -10 to 10 and y from -10 to 10, passing through points like (-10, -3), (0, 2), (10, 7) approximately)
select all the statements that are true.

• the y-intercept of function a is less than the y-intercept of function b.
• the y-intercept of function a is equal to the y-intercept of function b.
• the y-value of function a when x = 10 is less than the y-value of function b when x = 10.
• the y-value of function a when x = 10 is greater than the y-value of function b when x = 10.

Explanation:

Step1: Find slope of Function A

Using points \((-10, -2)\) and \((-5, -1)\), slope \(m_A = \frac{-1 - (-2)}{-5 - (-10)} = \frac{1}{5}\).
Equation: \(y - (-2) = \frac{1}{5}(x - (-10))\), simplify to \(y = \frac{1}{5}x\). So y - intercept of A is \(0\).

Step2: Find y - intercept of Function B

From the graph, Function B crosses y - axis at \(y = 2\). So y - intercept of B is \(2\).

Step3: Analyze y - intercepts

\(0 < 2\), so y - intercept of A is less than B. First statement is true, second is false.

Step4: Find y - value at \(x = 10\) for A

For Function A, \(y = \frac{1}{5}(10) = 2\).

Step5: Find y - value at \(x = 10\) for B

Function B: slope \(m_B=\frac{2 - (-3)}{0 - (-10)}=\frac{5}{10}=\frac{1}{2}\) (using points \((-10, -3)\) and \((0, 2)\)). Equation: \(y=\frac{1}{2}x + 2\). At \(x = 10\), \(y=\frac{1}{2}(10)+2 = 7\).

Step6: Compare y - values at \(x = 10\)

\(2 < 7\), so y - value of A at \(x = 10\) is less than B. Third statement is true, fourth is false.

Answer:

The true statements are:

• The \(y\)-intercept of Function A is less than the \(y\)-intercept of Function B.
• The \(y\)-value of Function A when \(x = 10\) is less than the \(y\)-value of Function B when \(x = 10\).