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Chapter 2: Problem 24

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$h(x)=x^{2}+16 x-17$$

Short Answer

Expert verified
The standard form of the function \(h(x)=x^{2}+16x-17\) is \(h(x)=(x+8)^{2}-81\). The vertex of the function is \(8, -81\), the axis of symmetry is at \(x = 8\), and \(x\)-intercepts are \(1, -17\).

Step by step solution

01

Convert the Quadratic Function into Standard Form

In order to convert the quadratic function into standard form, we need to complete the square. The standard form of a quadratic function is \(h(x)=a(x-h)^{2}+k\). So, we rewrite \(x^{2}+16x-17\) as \(x^{2}+16x+64-64-17\). This simplifies to \((x+8)^{2}-81\). Therefore, the function \(h(x)=x^{2}+16x-17\) in standard form is \(h(x)=(x+8)^{2}-81\).
02

Identify the Vertex

In the standard form, \((-h, k)\) is the vertex. So in our function \(h(x)=(x+8)^{2}-81\), the vertex is \((-(-8), -81) = (8, -81)\).
03

Identify the Axis of Symmetry

The axis of symmetry in a parabola \(y = ax^2+bx+c\) is \(x = -b/2a\). But in the vertex form, it's easier to identify it as being \(x = -h\). Hence, the axis of symmetry here is \(x = 8\).
04

Identify the \(x\)-intercepts

The \(x\)-intercepts are the values of \(x\) that make the function equal to zero. So, \(0 = (x+8)^{2}-81\) gives us \(x+8 = \pm \sqrt{81}\). Solving this provides \(x = -8 \pm 9\), that results in \(x_1 = 1\) and \(x_2 = -17\), which are the \(x\)-intercepts.

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