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Chapter 1: Problem 66

Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$y=9-x^{2}$$

Short Answer

Expert verified
X-intercepts: (3,0) and (-3,0). Y-intercept: (0,9). Symmetric about the y-axis.

Step by step solution

01

Establishing the Equation

The given equation is \( y = 9 - x^2 \). This is a quadratic equation that represents a downward-opening parabola.
02

Creating a Table of Values

Select a range of \( x \) values, calculate \( y \) for each to form coordinate pairs. Choose \( x = -3, -2, -1, 0, 1, 2, 3 \). This results in pairs: - \( (x = -3, y = 0) \), - \( (x = -2, y = 5) \), - \( (x = -1, y = 8) \), - \( (x = 0, y = 9) \), - \( (x = 1, y = 8) \), - \( (x = 2, y = 5) \), - \( (x = 3, y = 0) \). These values form the points which will be plotted to sketch the graph.
03

Plotting Points and Sketching the Graph

Plot the points on a graph, with \( x \) values on the x-axis and corresponding \( y \) values on the y-axis. Connect the points to form the parabola.
04

Finding the X-intercepts

X-intercepts occur when \( y = 0 \). Set the equation to \( 0 = 9 - x^2 \) and solve for \( x \). \( 0 = 9 - x^2 \) implies \( x^2 = 9 \), so \( x = \pm 3 \). The x-intercepts are at \( (3, 0) \) and \( (-3, 0) \).
05

Finding the Y-intercept

Y-intercepts occur when \( x = 0 \). Substitute \( x = 0 \) into \( y = 9 - x^2 \) to find \( y = 9 \). The y-intercept is at \( (0, 9) \).
06

Testing for Symmetry

The function is symmetric about the y-axis if replacing \( x \) with \( -x \) produces an equivalent expression. Evaluate \( y(-x) = 9 - (-x)^2 = 9 - x^2 \), which shows symmetry about the y-axis as it replicates the original equation.

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Key Concepts

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

Parabola
A parabola is a U-shaped curve that can open upwards or downwards, depending on the quadratic equation. In the equation given, \( y = 9 - x^{2} \), the parabola opens downwards because of the negative coefficient in front of \( x^{2} \).
  • The term \( x^{2} \) indicates that this is a quadratic function, with \( x \) being raised to the power of 2.
  • When graphed, each point of the equation \( (x, y) \) lies on the parabola.
  • In this case, the parabola reaches its highest point, or vertex, at the peak \( (0,9) \).
The vertex is crucial as it represents the maximum value of \( y \) for this downward-opening curve.
X-intercepts
The x-intercepts are the points where the parabola crosses the x-axis. These occur where the value of \( y \) is 0 in the equation \( y=9-x^2 \).
  • To find the x-intercepts, set \( y = 0 \) and solve for \( x \).
  • This results in the equation \( 0 = 9 - x^{2} \), which simplifies to \( x^{2} = 9 \).
  • Taking the square root of both sides gives two solutions: \( x = 3 \) and \( x = -3 \).
Thus, the x-intercepts for this parabola are \( (3,0) \) and \( (-3,0) \). These points are where the graph intersects the x-axis.
Y-intercept
The y-intercept is where the graph crosses the y-axis, signifying the value of \( y \) when \( x \) is zero. For the equation \( y = 9 - x^2 \):
  • Substitute \( x = 0 \) into the equation: \( y = 9 - (0)^2 \).
  • This simplifies to \( y = 9 \).
The y-intercept occurs at \( (0, 9) \).
This point can easily be identified on a graph and it provides a clear indication of where the curve starts when drawing the graph. The y-intercept is typically straightforward to calculate for any quadratic equation, simplifying initial graphing efforts.
Symmetry
In most parabolic equations, the concept of symmetry is about mirroring across the y-axis. For \( y = 9 - x^2 \), symmetry can be tested by replacing \( x \) with \( -x \) in the equation.
  • Modify the equation to calculate \( y(-x) \): \( y = 9 - (-x)^2 = 9 - x^2 \).
  • Observe that the equation remains unchanged, suggesting symmetry about the y-axis.
This means that for every point \( (x, y) \) on the parabola, there is an equivalent point \( (-x, y) \) on the other side of the y-axis. Symmetry in parabolas provides a helpful guide for accurately sketching them and understanding their behavior.

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