Question

if $f(x) = -x^2 + 3x + 5$ and $g(x) = x^2 + 2x$, which graph shows the graph of $(f + g)(x)$?

Explanation:

Step1: Define $(f+g)(x)$

$(f+g)(x) = f(x) + g(x)$

Step2: Substitute given functions

$(f+g)(x) = (-x^2 + 3x + 5) + (x^2 + 2x)$

Step3: Combine like terms

$(f+g)(x) = (-x^2 + x^2) + (3x + 2x) + 5 = 5x + 5$

Step4: Identify the line's key features

The function $y=5x+5$ is a linear equation with slope $m=5$ and y-intercept $(0,5)$. When $y=0$, $0=5x+5 \implies x=-1$, so x-intercept is $(-1,0)$.

Answer:

The correct graph is the first one shown: a straight line passing through $(0,5)$ and $(-1,0)$, with a positive steep slope matching $y=5x+5$.