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Chapter 2: Problem 26

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts. $$ y=-x^{2}+1 $$

Short Answer

Expert verified
The y-intercept is (0, 1). The x-intercepts are (1, 0) and (-1, 0).

Step by step solution

01

Find the y-intercept

The y-intercept occurs when x = 0. Substitute x = 0 into the equation: \( y = -x^2 + 1 \). This gives us \( y = -(0)^2 + 1 = 1 \). Thus, the y-intercept is (0, 1).
02

Find the x-intercepts

The x-intercepts occur when y = 0. Set the equation \( y = -x^2 + 1 \) equal to zero: \( 0 = -x^2 + 1 \). Solving for x: \( x^2 = 1 \), thus \( x = \pm 1 \). Therefore, the x-intercepts are (1, 0) and (-1, 0).
03

Plot the intercepts

Plot the intercepts on the graph. The points to plot are (0, 1), (1, 0), and (-1, 0).
04

Sketch the graph

Use the shape of the quadratic function, which is a parabola opening downwards (since the coefficient of \( x^2 \) is negative), to draw the graph. Connect the points smoothly to form the parabola.
05

Label the intercepts

Clearly label the plotted intercepts on the graph: (0, 1), (1, 0), and (-1, 0).

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Key Concepts

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

y-intercept
The y-intercept is the point where the graph crosses the y-axis, which means that at this point, the value of x is zero. To find the y-intercept of the quadratic equation given in our exercise, we substitute x = 0 into the equation. For the equation \( y = -x^2 + 1 \), by setting x to 0, we get: \( y = -(0)^2 + 1 = 1 \). This means our y-intercept is the point (0, 1). This is where the graph of the equation will cross the y-axis. It's essential to identify this point because it gives us an important reference for plotting the graph.
x-intercepts
The x-intercepts are points where the graph crosses the x-axis, meaning that y is zero at these points. To find the x-intercepts, we set y to zero and solve for x. Using our equation \( y = -x^2 + 1 \), we set y to 0: \( 0 = -x^2 + 1 \). Solving this, we get: \( x^2 = 1 \), which simplifies to \( x = 1 \) and \( x = -1 \). Thus, the x-intercepts are at the points (1, 0) and (-1, 0). These points are crucial for plotting the graph as they provide the locations where the graph will intersect the x-axis.
parabola
A quadratic equation like \( y = -x^2 + 1 \) graphs as a parabola. Parabolas can open upwards or downwards. The direction they open depends on the coefficient of the \( x^2 \) term:
  • If the coefficient is positive, the parabola opens upwards.
  • If the coefficient is negative, it opens downwards.
In our equation, the coefficient of \( x^2 \) is -1, which is negative. Therefore, the parabola opens downwards. The vertex of this parabola (the highest point) is at (0, 1), which is also the y-intercept.
plotting points
Plotting points is the process of placing points on the graph that satisfy the equation. For our quadratic equation, key points to plot include the intercepts we previously identified. We start by plotting (0, 1), the y-intercept. Then we plot (1, 0) and (-1, 0), the x-intercepts. To ensure a more accurate graph, we can find additional points by choosing values for x and solving for y. Since our parabola opens downwards, we can plot these points and sketch the curve by smoothly connecting them. The more points we plot, the easier it is to draw the correct shape of the parabola.

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