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Chapter 9: Problem 345

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it using properties. $$f(x)=2 x^{2}+4 x+6$$

Short Answer

Expert verified
The function in vertex form is \(f(x) = 2(x+1)^2 + 4\). The vertex is \((-1, 4)\) and the parabola opens upwards.

Step by step solution

01

- Identify coefficients

Identify the coefficients in the given quadratic function. The function given is: \(f(x) = 2x^2 + 4x + 6\). Here, \(a = 2\), \(b = 4\), and \(c = 6\).
02

- Complete the square

To rewrite the function in the form \(f(x) = a(x-h)^2 + k\), start by completing the square. First, factor the coefficient of \(x^2\) out of the \(x\) terms: \(f(x) = 2(x^2 + 2x) + 6\). Next, complete the square inside the parentheses. Take half of the coefficient of \(x\), square it, and add and subtract this inside the parentheses: \(2(x^2 + 2x + 1 - 1) + 6\). This simplifies to \(2((x+1)^2 - 1) + 6\). Finally, distribute and simplify the expression: \(f(x) = 2(x+1)^2 - 2 + 6 = 2(x+1)^2 + 4\).
03

- Rewrite the function

Write the function in the vertex form based on completing the square: \(f(x) = 2(x+1)^2 + 4\). Here, \(a = 2\), \(h = -1\), and \(k = 4\).
04

- Identify graph properties

Identify the properties of the function to graph it: The vertex of the parabola is at \((-1, 4)\). Since \(a = 2\) (positive), the parabola opens upwards. The axis of symmetry is \(x = -1\).
05

- Sketch the graph

Using the vertex \((-1, 4)\) and the fact that the parabola opens upwards, plot the vertex and a few points on either side of the axis of symmetry, ensuring to reflect the values symmetrically. Connect the points with a smooth parabolic curve.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vertex form
Understanding the vertex form of a quadratic function is crucial for easily identifying the graph's key features. The vertex form of a quadratic function is written as: \(f(x) = a(x-h)^2 + k\). In this form:
  • \
completing the square
Completing the square is a method used to convert a standard quadratic equation into vertex form. Here's the step-by-step process for completing the square:
  • Factor out the coefficient of \(x^2\) from the first two terms, if necessary.
  • Take half of the coefficient of \(x\), square it, and add and subtract this value inside the parentheses.
  • Simplify and rewrite the equation in the form \(a(x-h)^2 + k\).
For the function \(f(x) = 2x^2 + 4x + 6\), follow these steps to complete the square:
  • Factor out the coefficient of \(x^2\): \(2(x^2 + 2x) + 6\).
  • Take half of 2 (which is 1), square it (1), and add and subtract inside the parentheses: \(2(x^2 + 2x + 1 - 1) + 6\).
  • Rewrite as \(2((x+1)^2 - 1) + 6\) and simplify: \(2(x+1)^2 + 4\).
The function in vertex form is \(f(x) = 2(x+1)^2 + 4\).
graphing parabolas
Graphing parabolas involves identifying key features from the vertex form of the quadratic function. Here's how you can graph the parabola:
  • Identify the vertex \((h, k)\). For the function \(f(x) = 2(x+1)^2 + 4\), the vertex is \((-1, 4)\).
  • Determine whether the parabola opens upward or downward by looking at the coefficient \(a\). If \(a > 0\), it opens upward; if \(a < 0\), it opens downward. Since \(a = 2\), the parabola opens upwards.
  • Find the axis of symmetry, which is the line \(x = h\). In this case, the axis of symmetry is \(x = -1\).
Finally, plot the vertex and a few additional points on either side of the axis of symmetry, and connect them with a smooth curve.
axis of symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror images. It is essential for graphing the quadratic function accurately.
The axis of symmetry is given by the formula \(x = h\) when the quadratic function is in vertex form \(f(x) = a(x-h)^2 + k\).
In our example, after rewriting \(f(x) = 2(x+1)^2 + 4\), we identify the axis of symmetry as \(x = -1\).

When graphing, draw this line first. It will act as a guide, ensuring each plotted point on one side has a corresponding point on the other side. This symmetry helps in achieving an accurate and neat parabolic graph.

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Most popular questions from this chapter

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=-3 x^{2}+6 x+1$$

Graph each function using a horizontal shift. $$f(x)=(x-5)^{2}$$

Graph each function using a horizontal shift. $$f(x)=(x+5)^{2}$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=x^{2}+6 x+5$$

Find the intercepts of the parabola whose function is given. $$f(x)=-x^{2}+8 x-19$$
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