Problem 95 - Introductory Algebra for College Students

Question

Use a graphing utility to solve \((x-1)^{2}-9=0\) Graph \(y=(x-1)^{2}-9\) in a \([-5,5,1]\) by \([-9,3,1]\) viewing rectangle. The equation's solutions are the graph's \(x\) -intercepts. Check by substitution in the given equation.

Step-by-Step Solution

Answer

The solutions of the equation \((x-1)^{2}-9=0\) are \(x=4\) and \(x=-2\). The graph of the function \(y=(x-1)^{2}-9\) plotted in the viewing rectangle \([-5,5,1]\) by \([-9,3,1]\) confirms these solutions as they are the x-intercepts of the graph.

Step 1 - Solve the equation

First, solve the quadratic equation. \((x-1)^{2}-9=0\), to find the x-intercepts. The x intercepts are solutions for when \(y=0\). Reformulate equation as \((x-1)^{2}=9\). Then, solve for \(x\), which gives \(x=4\) and \(x=-2\).

Step 2 - Graph the equation

Use a graphing utility to plot \(y=(x-1)^{2}-9\). Place on the graph where \(y=0\), which gets \(x=4\) and \(x=-2\), these are the points where the graph crosses the x-axis.

Step 3 - Check solutions

Substitute \(x=4\) and \(x=-2\) back into the original equation to make sure they satisfy it. Substituting \(x=4\) gives \((4-1)^{2}-9=0\), and substituting \(x=-2\) gives \((-2-1)^{2}-9=0\), which are both correct.

Key Concepts

graphing utility

Graphing utilities are powerful tools that help in visualizing mathematical equations. These utilities can come in the form of software or calculator functions, and they allow you to plot graphs easily. When dealing with quadratic equations, like the one in our exercise, a graphing utility aids in understanding the behavior of the equation. You input the equation, \[ y=(x-1)^2-9 \] and observe how it is represented visually. Such visualizations help you see important points like where the curve intersects the x-axis, offering a clear representation of the solutions or roots.

• Make sure to adjust the viewing window to see the graph properly. In this case, use \([-5,5,1]\) by \([-9,3,1]\).
• Graphing utilities can also find exact values for intercepts and vertices efficiently, saving time and reducing errors.

x-intercepts

X-intercepts, also known as roots or zeros of the equation, are the points where the graph crosses the x-axis. This is where the value of \(y\) becomes zero. Determining x-intercepts is a crucial part of solving quadratic equations as they represent the real solutions of the equation.
In the quadratic equation from the exercise, \((x-1)^2-9=0\), you reformulate it to find when \(y=0\), leading to \(x-1)^2=9\). Solving this results in \(x=4\) and \(x=-2\).

• Check the calculated x-intercepts using your graphing utility by ensuring the graph passes through these points on the x-axis.
• This verification step ensures you've computed the intercepts correctly.

solving equations

To solve quadratic equations like \((x-1)^2-9=0\), it's essential to simplify them step by step.
First, convert the equation into a more workable form. Start by moving terms around: \((x-1)^2 = 9\).Remove the square by taking the square root on both sides, keeping in mind both positive and negative roots:

• The square root of \(9\) yields solutions \(x-1 = 3\) and \(x-1 = -3\).
• Solving these gives \(x = 4\) and \(x = -2\).

These are the solutions, or x-intercepts, for the quadratic equation. Double-check your work by substituting these values back into the original equation to ensure they satisfy it.

coordinate plane

The coordinate plane is a two-dimensional grid where you can plot equations like \(y=(x-1)^2-9\). Understanding this plane is crucial for visualizing mathematics.
It has an x-axis (horizontal) and a y-axis (vertical) where any point can be represented by \((x,y)\) coordinates.

• The equation given is graphed on this coordinate plane, making it easier to see where the parabola, or U-shaped curve, crosses the x-axis.
• A correctly set viewing rectangle, \([-5,5,1]\) for x-axis and \([-9,3,1]\) for y-axis, ensures that important features like intercepts and vertex are visible.

Visual representation on the coordinate plane solidifies understanding by showing how the algebraic solutions relate to the graph.