Question

which equation choice could represent the graph shown below?
answer
$f(x) = (x + 7)(x + 7)$
$f(x) = x(x + 7)$
$f(x) = x(x - 7)$
$f(x) = (x - 7)(x - 7)$

Explanation:

Step1: Find x-intercepts

The graph intersects the x - axis at \(x = 0\) and \(x=- 7\) (since it crosses the x - axis at the origin \((0,0)\) and at \(x=-7\)). For a quadratic function in factored form \(f(x)=(x - a)(x - b)\), the x - intercepts are at \(x = a\) and \(x = b\). If we rewrite the factored form to match our intercepts, when \(x = 0\), one factor is \(x\) (since \(x-0=x\)), and when \(x=-7\), the other factor is \((x + 7)\) (since \(x-(-7)=x + 7\)). So the function should be \(f(x)=x(x + 7)\).
Let's check the other options:

• For \(f(x)=(x + 7)(x + 7)\), the x - intercept is at \(x=-7\) (a double root), but the graph has two distinct x - intercepts (\(x = 0\) and \(x=-7\)), so this is incorrect.
• For \(f(x)=x(x - 7)\), the x - intercepts would be at \(x = 0\) and \(x = 7\), but the graph has an x - intercept at \(x=-7\) not \(x = 7\), so this is incorrect.
• For \(f(x)=(x - 7)(x - 7)\), the x - intercept is at \(x = 7\) (a double root), which does not match the graph's x - intercepts, so this is incorrect.

Answer:

\(f(x)=x(x + 7)\)