Question

which statements are true about the graph of the function $f(x) = x^2 - 8x + 5$? select three options.

• the function in vertex form is $f(x) = (x - 4)^2 - 11$.
• the vertex of the function is $(-8, 5)$.
• the axis of symmetry is $x = 5$.
• the $y$-intercept of the function is $(0, 5)$.
• the function crosses the $x$-axis twice.

Explanation:

Step1: Convert to vertex form

Complete the square for $f(x)=x^2-8x+5$:

$$\begin{align*} f(x)&=x^2-8x+16-16+5\\ &=(x-4)^2-11 \end{align*}$$

Step2: Identify vertex/axis of symmetry

From vertex form $f(x)=(x-h)^2+k$, vertex is $(h,k)=(4,-11)$, axis of symmetry is $x=4$.

Step3: Find y-intercept

Substitute $x=0$:
$f(0)=0^2-8(0)+5=5$, so y-intercept is $(0,5)$.

Step4: Check x-axis crossings

Calculate discriminant of $x^2-8x+5=0$:
$\Delta=b^2-4ac=(-8)^2-4(1)(5)=64-20=44>0$, so two real roots (crosses x-axis twice).

Answer:

A. The function in vertex form is $f(x) = (x - 4)^2 - 11$.
D. The y-intercept of the function is (0, 5).
E. The function crosses the x-axis twice.