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Magazine Chapter 1: Problem 61
Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$y=1-x^{2}$$
Short Answer
Expert verified
The x-intercepts are (1,0) and (-1,0), the y-intercept is (0,1), and the graph is symmetric about the y-axis.
Step by step solution
01
Make a Table of Values
Choose values for \(x\) and then calculate the corresponding \(y\) values using the equation \(y=1-x^{2}\). A good range for \(x\) would be from -3 to 3 for a thorough analysis.| \(x\) | \(y\) ||-----|------------|| -3 | \(1-(-3)^2 = -8\) || -2 | \(1-(-2)^2 = -3\) || -1 | \(1-(-1)^2 = 0\) || 0 | \(1-(0)^2 = 1\) || 1 | \(1-(1)^2 = 0\) || 2 | \(1-(2)^2 = -3\) || 3 | \(1-(3)^2 = -8\) |
02
Sketch the Graph using Table of Values
Plot the points from your table on the coordinate plane: (-3,-8), (-2,-3), (-1,0), (0,1), (1,0), (2,-3), (3,-8). Connect these points smoothly, forming a parabola that opens downward.
03
Find X-Intercepts
The x-intercepts occur where \(y = 0\). Substitute \(y = 0\) into the equation and solve: \[ 0 = 1 - x^2 \] Rearrange to solve for \(x\):\[ x^2 = 1 \]\[ x = \pm 1 \]The x-intercepts are at (1,0) and (-1,0).
04
Find Y-Intercept
The y-intercept occurs where \(x = 0\). Substitute \(x = 0\) into the equation:\[ y = 1 - (0)^2 = 1 \]The y-intercept is at (0,1).
05
Test for Symmetry
To test for symmetry, check for three types: about the x-axis, y-axis, and the origin.- **X-axis Symmetry:** Replace \(y\) with \(-y\): -y = 1 - x^2. This equation is not equivalent to the original, indicating no symmetry about the x-axis.- **Y-axis Symmetry:** Replace \(x\) with \(-x\): y = 1 - (-x)^2 = 1 - x^2. This is the same as the original equation, indicating symmetry about the y-axis.- **Origin Symmetry:** Replace \(x\) with \(-x\) and \(y\) with \(-y\): -y = 1 - (-x)^2 = 1 - x^2. This is not equivalent to the original equation, indicating no symmetry about the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding X-Intercepts in Parabolas
An x-intercept is a point where the graph of an equation crosses the x-axis. At this point, the y-value is zero because it lies on the x-axis. To find the x-intercepts for the equation \( y = 1 - x^2 \), we set \( y \) to zero and solve for \( x \).
This gives us:
This gives us:
- \( 0 = 1 - x^2 \)
- \( x^2 = 1 \)
- \( x = \pm 1 \)
Decoding Y-Intercepts in Graphs
The y-intercept of a graph is the point where the line or curve crosses the y-axis. At this point, the x-value is zero. For the equation \( y = 1 - x^2 \), the y-intercept can be found by substituting \( x = 0 \) into the equation.
So you get:
So you get:
- \( y = 1 - (0)^2 \)
- \( y = 1 \)
Exploring Axis Symmetry in Parabolic Graphs
Symmetry in graphs simplifies the graphing process and reveals important characteristics of functions. The three common types of symmetry are concerning the x-axis, y-axis, and the origin. Let's see how this applies to our equation \( y = 1 - x^2 \):
- X-axis Symmetry: If an equation is symmetric about the x-axis, replacing \( y \) with \( -y \) should yield the original equation. Here, \(-y = 1 - x^2\) does not equate to \(y = 1 - x^2\), so no x-axis symmetry.
- Y-axis Symmetry: When \( x \) is replaced with \( -x \), the equation \( y = 1 - (-x)^2 \) simplifies to \( y = 1 - x^2 \), the same as the original. This indicates symmetry about the y-axis.
- Origin Symmetry: Substituting \( x \) with \( -x \) and \( y \) with \( -y \), we get \(-y = 1 - x^2\) which is not the original equation, so no symmetry about the origin.
Graphing Equations of Parabolas
Graphing an equation like \( y = 1 - x^2 \) involves plotting several points to visualize the curve. A table of values can aid in selecting these points effectively. Here's how you can proceed:
First, choose a set of x-values: a common range is from \(-3\) to \(3\). Calculate the corresponding y-values using the equation. For example, if \( x = -3 \), then \( y = 1 - (-3)^2 = -8 \). Continue this process for each x-value.
As you plot points like \((0, 1), (-1, 0), (1, 0), (-2, -3)\), etc., connect them smoothly to form the parabola.
First, choose a set of x-values: a common range is from \(-3\) to \(3\). Calculate the corresponding y-values using the equation. For example, if \( x = -3 \), then \( y = 1 - (-3)^2 = -8 \). Continue this process for each x-value.
As you plot points like \((0, 1), (-1, 0), (1, 0), (-2, -3)\), etc., connect them smoothly to form the parabola.
- The shape of a parabola in this form (opening downwards) arises from the negative coefficient of \( x^2 \).
- The vertex, the highest point, is at \( (0, 1) \), the y-intercept, for this downward-opening parabola.
