Question

use a graphing utility to graph the quadratic function.
$f(x) = 2x^2 - 24x + 72$
identify the vertex, axis of symmetry, and $x$-intercept(s). then check your results algebraically by writing the quadratic function in standard form. (if not exist, enter dne.)
vertex
$(x, f(x)) = (\square)$
axis of symmetry
$\square$
$x$-intercept
$(x, f(x)) = (\square)$
standard form
$f(x) = \square$

Explanation:

Step1: Rewrite the function in standard form (vertex form)

We start with the quadratic function \( f(x) = 2x^2 - 24x + 72 \). First, factor out the coefficient of \( x^2 \) from the first two terms:
\( f(x) = 2(x^2 - 12x) + 72 \)
To complete the square inside the parentheses, we take half of the coefficient of \( x \) (which is \( -12 \)), square it, and add and subtract it inside the parentheses. Half of \( -12 \) is \( -6 \), and \( (-6)^2 = 36 \). So we have:
\( f(x) = 2(x^2 - 12x + 36 - 36) + 72 \)
Now, we can rewrite the expression inside the parentheses as a perfect square trinomial and a constant:
\( f(x) = 2((x - 6)^2 - 36) + 72 \)
Distribute the 2:
\( f(x) = 2(x - 6)^2 - 72 + 72 \)
Simplify the constants:
\( f(x) = 2(x - 6)^2 \)

Step2: Identify the vertex

The standard form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where the vertex is \( (h, k) \). From our standard form \( f(x) = 2(x - 6)^2 + 0 \), we can see that \( h = 6 \) and \( k = 0 \). So the vertex is \( (6, 0) \).

Step3: Identify the axis of symmetry

The axis of symmetry of a quadratic function in the form \( f(x) = a(x - h)^2 + k \) is the vertical line \( x = h \). Since \( h = 6 \), the axis of symmetry is \( x = 6 \).

Step4: Find the x-intercept(s)

To find the x-intercepts, we set \( f(x) = 0 \) and solve for \( x \):
\( 2(x - 6)^2 = 0 \)
Divide both sides by 2:
\( (x - 6)^2 = 0 \)
Take the square root of both sides:
\( x - 6 = 0 \)
So \( x = 6 \). Therefore, the x-intercept is \( (6, 0) \) (a repeated root, so it's a single x-intercept at \( x = 6 \)).

Step5: Confirm the standard form

From Step1, we found that the standard form of the function is \( f(x) = 2(x - 6)^2 \).

Answer:

• vertex: \( (6, 0) \)
• axis of symmetry: \( x = 6 \)
• x-intercept: \( (6, 0) \)
• standard form: \( f(x) = 2(x - 6)^2 \)