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Chapter 1: Problem 58

Use a graphing calculator to graph the half-circle \(y=\sqrt{25-(x-4)^{2}}\). Then, use the INTERCEPT feature to find the value of both the \(x\) - and \(y\) -intercepts.

Short Answer

Expert verified
The x-intercepts are at x = -1 and x = 9, and the y-intercept is at y = 3.

Step by step solution

01

Understanding the Equation

The given equation is of the form \( y = \sqrt{r^2 - (x-h)^2} \), which represents the upper half of a circle centered at \((h, 0)\) with radius \( r \). In this case, \( h = 4 \) and \( r = 5 \), so the circle is centered at \((4, 0)\) with a radius of 5 units.
02

Graphing the Half-Circle

To graph the half-circle using a graphing calculator, input the equation \( y = \sqrt{25-(x-4)^{2}} \). This will plot the upper half of the circle with a center at (4, 0) and a radius of 5.
03

Finding the x-intercepts

The x-intercepts occur where \( y = 0 \). Set \( \sqrt{25-(x-4)^2} = 0 \) and solve for \( x \). This gives \( 25 = (x-4)^2 \), leading to \( x-4 = \pm 5 \). Therefore, the x-intercepts are at \( x = 9 \) and \( x = -1 \).
04

Finding the y-intercept

The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the equation: \( y = \sqrt{25-(0-4)^2} = \sqrt{25-16} = \sqrt{9} = 3 \). Thus, the y-intercept is at \( y = 3 \).
05

Using the INTERCEPT Feature

Use the INTERCEPT feature on your graphing calculator to confirm the points where the graph intersects the axes. This will provide you with the \( x \)-intercepts at \( x = -1 \) and \( x = 9 \), and the \( y \)-intercept at \( y = 3 \).

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Key Concepts

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

Half-Circle Equation
The half-circle equation is a mathematical expression that represents one side (usually the upper side) of a circle. The given equation, \( y = \sqrt{25-(x-4)^2} \), is structured in a way that it plots the top half of a circle. This equation can be written in the general form \( y = \sqrt{r^2 - (x-h)^2} \). Here, \( (h, 0) \) is the center of the circle and \( r \) is the radius.
For our equation:
  • Center, \( h = 4 \)
  • Radius, \( r = 5 \)
This means the circle has a center at the point (4, 0) on the graph, and stretches 5 units away from this center, forming a perfect half-circle above the x-axis.
x-intercepts
The x-intercepts of a graph are points where the graph cuts through the x-axis. These occur when \( y = 0 \). In a half-circle equation, you find these intercepts by setting the equation \( y = \sqrt{r^2 - (x-h)^2} = 0 \).
For the given equation, you solve \( \sqrt{25-(x-4)^2} = 0 \), leading to:
  • \( 25 = (x-4)^2 \)
  • \( x-4 = \pm 5 \)
  • Solve for \( x \): \( x = 9 \) and \( x = -1 \)
These points, \( x = 9 \) and \( x = -1 \), represent the x-coordinates where the half-circle curve meets the x-axis. This tells us that the half-circle touches the x-axis at these positions.
y-intercepts
The y-intercept is where the graph meets or crosses the y-axis. To find this intercept in a half-circle equation, you set \( x = 0 \) and solve for \( y \). This helps you determine where the graph touches the y-axis.
Let's do the calculation for the given equation:
  • Substitute \( x = 0 \) into \( y = \sqrt{25-(0-4)^2} \)
  • Compute: \( y = \sqrt{25-16} = \sqrt{9} = 3 \)
So, the y-intercept here is \( y = 3 \). This point shows that the top half of the circle touches the y-axis at 3 units up the vertical line from the origin.
Graphing Functions
Graphing functions is an essential skill in both mathematics and scientific analyses, allowing for clear visualization of equations or expressions. When it comes to graphing a half-circle like \( y = \sqrt{25-(x-4)^{2}} \), a graphing calculator becomes particularly useful. Here, you can quickly input the equation and see its graphical representation.
Steps you can follow with a graphing calculator:
  • Enter the equation into the calculator's graphing feature.
  • Observe the half-circle plot, extending from \((x = -1)\) to \((x = 9)\).
  • Use features like "zoom" to understand the curve's details better.
  • Employ the "INTERCEPT" function to confirm x- and y-intercepts found analytically.
By mastering these steps, you get a visual confirmation of the theoretical calculations you perform by hand, making graphing functions a powerful tool in comprehending geometric shapes such as circles and their sections.

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