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Chapter 9: Problem 41

Graph each parabola. Give the vertex, axis of symmetry, domain, and range. $$ f(x)=-2(x+3)^{2}+4$$

Short Answer

Expert verified
Vertex: \((-3, 4)\), Axis: \(x = -3\), Domain: \((-\infty, \infty)\), Range: \((-\infty, 4]\).

Step by step solution

01

Identify the Standard Form of the Equation

The given equation is in the vertex form of a parabola, which can be written as: \[ f(x) = a(x-h)^2 + k \] Here, \(a = -2\), \(h = -3\), and \(k = 4\).
02

Find the Vertex

The vertex \((h,k)\) can be directly read from the equation. Therefore, the vertex is \((-3, 4)\).
03

Determine the Axis of Symmetry

The axis of symmetry in a parabola given in vertex form is the vertical line that passes through the vertex. Hence, the axis of symmetry is \(x = -3\).
04

Determine the Domain

For any quadratic function, the domain includes all real numbers since parabolas extend infinitely in both the left and right directions. Therefore, the domain is \( (-\infty, \infty) \).
05

Determine the Range

Since the coefficient \(a = -2\) is negative, the parabola opens downwards. The maximum value of the parabola is the y-coordinate of the vertex. Since the vertex is \((-3, 4)\), the maximum y-value is 4. Therefore, the range is \( (-\infty, 4] \).
06

Graph the Parabola

Plot the vertex \((-3, 4)\), and draw the axis of symmetry \(x = -3\). Since \(a = -2\) indicates the parabola opens downward and is more narrow than the standard parabola. Plot additional points by choosing x-values and solving for y, and then sketch the curve.

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Key Concepts

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

Vertex form of a parabola
The vertex form of a parabolic equation is extremely useful for identifying key features of the graph. It's written as: \[ f(x) = a(x-h)^2 + k \] where
  • a is the coefficient that determines the width and the direction of the parabola,
  • h is the x-coordinate of the vertex, and
  • k is the y-coordinate of the vertex.
In our equation \( f(x) = -2(x + 3)^2 + 4 \), we can see that \( a = -2 \), \( h = -3 \), and k = 4. This directly tells us the vertex of the parabola is at (-3, 4), and because \( a \) is negative, the parabola opens downwards.
Axis of symmetry
The axis of symmetry is a vertical line that passes through the vertex of the parabola. This line divides the parabola into two mirror-image halves. In the vertex form \( f(x) = a(x-h)^2 + k \), the axis of symmetry can be determined directly from the vertex. Here, because our vertex is at (-3, 4), the axis of symmetry is \( x = -3 \). When you draw the graph, you can use this line to help ensure your parabola is symmetric.
Domain and range
Understanding the domain and range of a quadratic function is essential. For any quadratic function, the domain is all real numbers. This is because parabolas continue infinitely left and right. Hence, the domain is \( (-\infty, \infty) \). The range depends on whether the parabola opens upwards or downwards. In the given equation \( f(x) = -2(x + 3)^2 + 4 \), because \( a = -2 \) is negative, the parabola opens downwards. This means that the highest point on the graph is the vertex. Therefore, the range is all values less than or equal to the y-coordinate of the vertex, which is 4. Thus, the range of the given function is \( (-\infty, 4] \).
Quadratic function
A quadratic function takes the form \( f(x) = ax^2 + bx + c \) and its graph is a parabola. The standard and vertex forms of a quadratic function provide different insights into its properties.
  • The standard form \( ax^2 + bx + c \) is useful for finding the y-intercept and using the quadratic formula to find the roots.
  • The vertex form \( a(x-h)^2 + k \) is useful for identifying the vertex and the direction in which the parabola opens (upward if \( a > 0 \); downward if \( a < 0 \)).
Quadratic functions are fundamental to algebra, showing up in various contexts including projectile motion and optimization problems.
Graphing quadratic equations
To graph a quadratic equation, follow these steps:
  • Identify the form of your quadratic equation (standard or vertex form).
  • Find the vertex of the parabola.
  • Determine the axis of symmetry.
  • Plot the vertex and draw the axis of symmetry.
  • Pick additional points on both sides of the axis of symmetry, substitute x-values into the function to find corresponding y-values, and plot these points.
  • Draw a smooth curve through the points making sure it is symmetric about the axis of symmetry.
For the given function \( f(x) = -2(x + 3)^2 + 4 \), start by plotting the vertex (-3, 4). Draw the axis of symmetry at \( x = -3 \). Then choose some x-values around -3, compute their corresponding y-values, plot these points, and sketch the downward-opening curve.

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Most popular questions from this chapter

For each pair of functions \(f\) and \(g\), find \((a) f+g,\) (b) \(f-g,\) (c) \(f g\), and \((d) \frac{f}{g}\). Give the domain for each. See Example 2. $$ f(x)=4 x-1, \quad g(x)=6 x+3 $$

Use the tests for symmetry to decide whether the graph of each relation is symmetric with respect to the \(x\) -axis, the y-axis, or the origin. More than one of these symmetries, or none of them, may apply. $$ x y=2 $$

Graph each piecewise function. \(f(x)=\left\\{\begin{array}{ll}2+x & \text { if } x<-4 \\ -x^{2} & \text { if } x \geq-4\end{array}\right.\)

Sketch each graph. $$ g(x)=4 \sqrt{x} $$

For each pair of functions \(f\) and \(g\), find \((a) f+g,\) (b) \(f-g,\) (c) \(f g\), and \((d) \frac{f}{g}\). Give the domain for each. See Example 2. $$ f(x)=2 x+5, \quad g(x)=\sqrt{4 x+3} $$
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