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Chapter 2: Problem 61

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=1-x^{2}\)

Short Answer

Expert verified
The graph of the equation \(y=1-x^2\) is a downward opening parabola with its vertex at (0, 1). The intercepts are at points (1, 0), (-1, 0), and (0, 1). The graph is symmetrical about the y-axis.

Step by step solution

01

Identify the Vertex

The equation given is \(y = 1 - x^2\), which is already in the vertex form, comparable to \(y=a(x-h)^2 + k\), where a is -1, h is 0 and k is 1. Thus, the vertex of the parabola is at the point (0, 1).
02

Find the Intercepts

For x-intercepts, set \(y=0\) and solve for \(x\). The equation will be \(0 = 1 - x^2\), which gives \(x^2 = 1\) and therefore \(x = 卤1\). The x-intercepts are (1, 0) and (-1, 0). The y-intercept is found by setting \(x=0\) in the equation. This provides \(y = 1 - 0 = 1\), so the y-intercept is at (0, 1).
03

Check for Symmetry

Identify if the graph is symmetrical by checking what changing \(x\) to \(-x\) in the equation does. \(y = 1 - (-x)^2 = 1 - x^2\), which is symmetrical around the y-axis (since \(x = -x\)). The graph of this equation is symmetrical.
04

Sketch the Graph

Begin by marking the vertex (0, 1) and the intercepts (1, 0), (-1, 0), and (0, 1) on the graph. The parabola opens downward since the coefficient of \(x^2\) is negative. Draw a smooth, curved line through these points to complete the parabolic graph.

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Key Concepts

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

Vertex Identification
Understanding the vertex of a parabola is crucial because it represents the highest or lowest point of the graph. In the equation \(y = 1 - x^2\), we can relate it to the vertex form \(y = a(x-h)^2 + k\), where \(h\) and \(k\) are the coordinates of the vertex.
Here, the equation is already simplified such that \(h = 0\) and \(k = 1\), resulting in a vertex at the point \((0, 1)\).
This vertex tells us that the graph will reach its peak at this point, as the coefficient of \(x^2\) (\(a = -1\)) is negative, indicating the parabola opens downward.
Intercepts Calculation
Calculating intercepts helps in determining where the graph crosses the axes.
  • X-Intercepts: To find the x-intercepts, set \(y = 0\) and solve the equation \(0 = 1 - x^2\).
    This results in \(x^2 = 1\), giving solutions \(x = 1\) and \(x = -1\).
    Thus, the x-intercepts are at the points \((1, 0)\) and \((-1, 0)\).
  • Y-Intercept: Set \(x = 0\) to find the y-intercept. Substituting gives \(y = 1 - 0 = 1\). Thus, the y-intercept is \((0, 1)\).

These intercepts are critical points that help provide a skeletal framework for graphing the equation.
Symmetry Testing
Testing a graph for symmetry can reveal if it mirrors on a particular axis. For the equation \(y = 1 - x^2\), replace \(x\) with \(-x\) to test for y-axis symmetry.
When substituted, \(y = 1 - (-x)^2\) simplifies to \(y = 1 - x^2\).
Since the equation remains unchanged, the graph is symmetrical about the y-axis.
This form of symmetry is typical of parabolas that open upward or downward, as they naturally mirror evenly across the y-axis.
Graph Sketching
With all the properties identified, you can now sketch the graph effectively.
  • Start by plotting the vertex \((0, 1)\), which is the top point here since the parabola opens downward.
  • Plot the intercepts: \((1, 0)\) and \((-1, 0)\) on the x-axis, and \((0, 1)\) on the y-axis.
  • Notice the downward opening direction, confirmed by the negative \(x^2\) coefficient.

Finally, draw a smooth curve through these points, ensuring the curve is symmetrical across the y-axis. This belly-down parabola will extend infinitely downward, growing wider but never touching any other point on the axes besides those identified.

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Most popular questions from this chapter

The weekly cost \(C\) of producing \(x\) units in a manufacturing process is given by the function \(C(x)=50 x+495\) The number of units \(x\) produced in \(t\) hours is given by \(x(t)=30 t\) Find and interpret \((C \circ x)(t)\).

Show that \(f\) and \(g\) are inverse functions by (a) using the definition of inverse functions and (b) graphing the functions. Make sure you test a few points, as shown in Examples 6 and 7 . \(f(x)=x^{3}, \quad g(x)=\sqrt[3]{x}\)

Consider the graph of \(g(x)=\sqrt{x}\) Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(g\) is reflected in the \(x\) -axis, shifted two units to the left, and shifted one unit upward.

Use a graphing utility to graph the six functions below in the same viewing window. Describe any similarities and differences you observe among the graphs. (a) \(y=x\) (b) \(y=x^{2}\) (c) \(y=x^{3}\) (d) \(y=x^{4}\) (e) \(y=x^{5}\) (f) \(y=x^{6}\)

Find (a) \((f+g)(x)\), (b) \((f-g)(x)\), (c) \((f g)(x)\), and (d) \((f / g)(x)\). What is the domain of \(f / g\) ? \(f(x)=x^{2}, g(x)=1-x\)
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