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Chapter 9: Problem 32

Graph each parabola. Give the vertex, axis of symmetry, domain, and range. \(f(x)=-2 x^{2}+12 x-13\)

Short Answer

Expert verified
Vertex: (3, 5). Axis of symmetry: x = 3. Domain: (-鈭�, 鈭�). Range: (-鈭�, 5].

Step by step solution

01

Find the Vertex

To find the vertex of the parabola given by the function \(f(x)=-2x^2+12x-13\), use the vertex formula \(x = -\frac{b}{2a}\). Here, \(a = -2\) and \(b = 12\). Compute: \(x = -\frac{12}{2(-2)} = 3\). Substitute \(x = 3\) back into the function to find \(y\): \(f(3) = -2(3)^2 + 12(3) - 13 = -18 + 36 - 13 = 5\). Therefore, the vertex is \((3, 5)\).
02

Determine the Axis of Symmetry

The axis of symmetry for the parabola is the vertical line that passes through the vertex. For this parabola, it is \(x = 3\).
03

Identify the Domain

The domain of any quadratic function is all real numbers. Therefore, the domain is \((-\text{infty}, \text{infty})\).
04

Identify the Range

Since the coefficient of \(x^2\) is negative (\(-2\)), the parabola opens downwards. The maximum value of the function is at the vertex (3, 5). The range is all real numbers that are less than or equal to 5: \((-\text{infty}, 5]\).
05

Graph the Parabola

Plot the vertex \((3, 5)\) on the graph. Since the parabola opens downwards, sketch the curve, ensuring it passes through key points. For example, compute function values at additional points (such as x = 2 or x = 4) to help shape the curve accurately.

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Key Concepts

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

Vertex of a Parabola
The vertex of a parabola is a significant point where the curve changes direction. Consider the function we are working with: \(f(x) = -2x^2 + 12x - 13\). The vertex can be found using the formula \(x = -\frac{b}{2a}\). Here, \(a = -2\) and \(b = 12\).

Plug these values into the formula to get: \(x = -\frac{12}{2(-2)} = 3\).

To determine the y-coordinate of the vertex, substitute \(x = 3\) back into the function:

\(f(3) = -2(3)^2 + 12(3) - 13 = -18 + 36 - 13 = 5\).

Therefore, the vertex is \((3, 5)\). The vertex tells us the peak point of the parabola when it opens downwards, like in this case.
Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex of the parabola. It divides the parabola into two mirror-image halves.

For the function \(f(x) = -2x^2 + 12x - 13\), we determined that the vertex is at \((3, 5)\).

Hence, the axis of symmetry is the line \(x = 3\).

This means for every point on the parabola, there is a corresponding point equidistant from the same line but on the other side. This line of symmetry helps when graphing the parabola as it provides a guide for plotting reflected points.
Domain and Range
The domain and range of a quadratic function are crucial concepts. Let's break them down:

  • Domain: The domain of any quadratic function is all real numbers (\( \mathbb{R} \)). This is because you can plug any real number into the function and get a valid output. Therefore, the domain for \(f(x) = -2x^2 + 12x - 13\) is \(( -\infty, \infty )\).
  • Range: The range depends on the direction the parabola opens. Since the coefficient of \(x^2\) is negative (\( -2 \)), the parabola opens downwards. The maximum value of the function (or the highest point on the graph) is at the vertex. From our calculations, the vertex is at \((3, 5)\). Hence, the y-values will be less than or equal to 5. Therefore, the range is \(( -\infty, 5 ]\).

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