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Chapter 2: Problem 21

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$f(x)=x^{2}+8 x+13$$

Short Answer

Expert verified
The standard form of the function is \(f(x) = (x+2)^2 + 9\). The vertex is at \((-2, 9)\) and the axis of symmetry is \(x = -2\). The function has no x-intercepts. The graph of the function is a parabola opening upwards, with vertex at \((-2, 9)\), symmetric about the line \(x = -2\).

Step by step solution

01

Rewrite the quadratic function in standard form

First, let's rewrite the quadratic function in standard form by completing the square. The given function \(f(x) = x^2 + 8x + 13\). Split the middle term 8x into 2 parts - 4x and 4x - while maintaining its value. So, you can rewrite as \(f(x) = (x^2 + 4x + 4) + 4x + 9\). Notice that \((x^2 + 4x + 4)\) is a perfect square trinomial. It can be written as \((x+2)^2\). Thus, the function in standard form is \(f(x) = (x+2)^2 + 9\)
02

Identify the vertex and axis of symmetry

For the function in standard form \(f(x) = a(x-h)^2 + k\), the vertex is at \((h, k)\). So in our function \(f(x) = (x+2)^2 + 9\), the vertex is at \((-2, 9)\). The axis of symmetry is the vertical line passing through the vertex, which is \(x = h\). Hence, the axis of symmetry in this case is \(x = -2\)
03

Identify x-intercepts

The x-intercepts are the x-values where \(f(x) = 0\). We need to solve the equation \((x+2)^2 + 9 = 0\). Because squaring any real value gives a result greater than or equal to 0 and adding 9 to it will yield a resultant larger than 0, the function has no x-intercepts.
04

Sketch the graph

Plot the vertex, which is the peak of the parabola at \((-2, 9)\). Since the co-efficient of \(x^2\) is positive, the parabola opens upwards. As the equation has no real x-intercepts, so the the parabola does not cross the x-axis. The parabola symmetrically opens upwards about the line \(x = -2\). For a precise sketch, we can plot additional points, though.

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Most popular questions from this chapter

Complete the following. \begin{aligned} &i^{1}=i \quad i^{2}=-1 \quad i^{3}=-i \quad i^{4}=1\\\ &i^{5}=\quad i^{6}=\quad i^{7}=\quad i^{8}=\\\ &i^{9}=\\\ &i^{10}=\quad i^{11}=\quad i^{12}= \end{aligned} What pattern do you see? Write a brief description of how you would find \(i\) raised to any positive integer power.

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=16 x^{3}-20 x^{2}-4 x+15$$

Geometry You want to make an open box from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. (a) Let \(x\) represent the side length of each of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume \(V\) of the box as a function of \(x .\) Determine the domain of the function. (c) Sketch the graph of the function and approximate the dimensions of the box that will yield a maximum volume. (d) Find values of \(x\) such that \(V=56 .\) Which of these values is a physical impossibility in the construction of the box? Explain.

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{1}{4} x^{2}-x+\frac{1}{4}=\frac{1}{4}\left(4 x^{3}-x^{2}-4 x+1\right)$$

Find the rational zeros of the polynomial function. $$P(x)=x^{4}-\frac{25}{4} x^{2}+9=\frac{1}{4}\left(4 x^{4}-25 x^{2}+36\right)$$
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