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Chapter 2: Problem 23

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$f(x)=x^{2}-14 x+54$$

Short Answer

Expert verified
The vertex form of the function is \(f(x) = (x - 7)^{2} + 5\), the vertex is at the point (7, 5), the axis of symmetry is the line \(x=7\), and there are no x-intercepts.

Step by step solution

01

Rewrite in Vertex (Standard) Form

First, the quadratic function \(f(x) = x^2 - 14x + 54\) needs to be written in the standard form. Remember that standard form is given by \(f(x) = a(x - h)^{2} + k\), where (h,k) are the coordinates of the vertex. Thus, \(f(x)\) can be rewritten as \(f(x) = (x - 7)^{2} + 5\). This comes from completing the square of the equation.
02

Identify the vertex and the axis of symmetry

The vertex of the quadratic function in its standard form can be found directly from the equation. Here, the vertex is at the point (7, 5). The axis of symmetry for a quadratic function is always a vertical line that goes through the vertex, thus the axis of symmetry is the line \(x = 7\).
03

Identify the x - intercept(s)

The x-intercepts of the function are found by setting \(f(x)\) to zero and solving for \(x\). Thus, \(0 = (x - 7)^{2} + 5\). Solving for x gives \(x = 7 \pm \sqrt{-5}\). However, since we cannot take the square root of a negative number, there are no x-intercepts for this function.

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