Question

function p is a linear function with a y-intercept of 6. function q is defined by the equation $y = -\frac{1}{2}x + 5$.

which statement must be true about functions p and q?
○ both functions have the same slope.
○ both functions have a negative slope.
○ the functions will have the same input when y = 6.
○ the functions will have different outputs when x = 0.

Explanation:

Step1: Recall linear function form

A linear function is $y=mx+b$, where $m$=slope, $b$=y-intercept.

Step2: Analyze Function P

Function P has $b=6$, so at $x=0$, $y=6$. Its slope $m_P$ is unknown.

Step3: Analyze Function Q

Function Q: $y=-\frac{1}{2}x+5$. At $x=0$, $y=5$. Its slope $m_Q=-\frac{1}{2}$.

Step4: Evaluate each option

• Option1: $m_P$ is unknown, can't confirm equal slopes.
• Option2: $m_P$ could be positive/zero, not guaranteed negative.
• Option3: For Q, set $y=6$: $6=-\frac{1}{2}x+5 \implies x=-2$. For P, $y=6$ at $x=0$ (since y-intercept=6). Inputs differ.
• Option4: At $x=0$, P outputs 6, Q outputs 5. Outputs are different.

Answer:

D. The functions will have different outputs when x = 0.