Problem 51 - College Algebra with Corequisite Support

Question

For the following exercises, use your graphing calculator to input the linear graphs in the \(Y=\) graph menu. After graphing it, use the \(2^{\text {nd }}\) CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see "left bound?" and a blinking cursor on the graph of the line. Move this cursor to the left of the \(x\) -intercept, hit ENTER. Now it says "right bound?" Move the cursor to the right of the x-intercept, hit ENTER. Now it says "guess?" Move your cursor to the left somewhere in between the left and right bound near the \(x\) -intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the \(x\) -intercept or the "zero" to the \(y\) -value. Use this to find the \(x\) -intercept. Note: With linear/straight line functions the zero is not really a "guess," but it is necessary to enter a "guess" so it will search and find the exact \(x\) -intercept between your right and left boundaries. With other types of functions (more than one \(x\) -intercept), they may be irrational numbers so "guess" is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries. $$ \mathrm{Y}_{1}=-8 x+6 $$

Step-by-Step Solution

Answer

The x-intercept is at \(x = 0.75\).

Step 1: Enter the Equation

Turn on your graphing calculator and press the 'Y=' button to access the equation entry menu. Enter the equation \( Y_1 = -8x + 6 \) into the first line.

Step 2: Graph the Line

After entering the equation, press the 'GRAPH' button. The calculator will display the graph of the line \( Y_1 = -8x + 6 \). Observe where the line crosses the x-axis.

Step 3: Access Calculator Menu

Press the '2nd' button followed by the 'TRACE' button to access the CALC menu. Select option '2: zero' from the menu to find the x-intercept of the graph.

Step 4: Set Left Bound

Use the left/right arrow keys to move the cursor on the graph to a point left of the x-intercept. Once the cursor is to the left, press 'ENTER'. This sets your left boundary.

Step 5: Set Right Bound

Now move the cursor to a point right of the x-intercept using the arrow keys and press 'ENTER'. This sets your right boundary.

Step 6: Guess the Zero

Move the cursor close to where the line crosses the x-axis, between the set boundaries, and press 'ENTER'. This is your guess of the x-intercept position.

Step 7: Find and Record the Intercept

After pressing 'ENTER', the calculator will display the x-intercept at the bottom of the screen. It should show the coordinate where the graph crosses the x-axis, typically as \((x, 0)\).

Key Concepts

Linear Equations

Linear equations are mathematical statements that show a straight-line relationship between two variables, usually called x and y. They are one of the simplest forms of equations and have broad applications in modeling relationships in various fields. A linear equation can be expressed in the form: \[ y = mx + b \]where:

• \( m \) is the slope of the line, indicating its steepness and direction.
• \( b \) is the y-intercept, where the line crosses the y-axis.

The equation you entered into your graphing calculator, \( Y_1 = -8x + 6 \), is an example of such a linear equation. Here, \( m = -8 \) suggests that the line goes downhill from left to right, and the y-intercept is at \( 6 \), meaning it crosses the y-axis at the point \((0, 6)\).

X-Intercept

The x-intercept is the point where a graph crosses the x-axis on a coordinate plane. In the context of linear equations, finding this intercept is crucial because it provides valuable information about the factors of the equation and where the function reaches zero. To find the x-intercept, you set \( y = 0 \) in the equation and solve for x. For the equation in our exercise, \( Y_1 = -8x + 6 \), setting \( y = 0 \) gives:\[ 0 = -8x + 6 \]By solving for x, you find:\[ 8x = 6 \]\[ x = \frac{3}{4} \]This solution shows that the graph crosses the x-axis at \( x = \frac{3}{4} \).Understanding how to find the x-intercept helps in analyzing the behavior of linear equations and their respective graphs.

Graphing Techniques

Graphing is a powerful visual tool that helps in understanding mathematical concepts. When graphing linear equations like \( Y_1 = -8x + 6 \) on a calculator, there are several techniques to improve your understanding:

• **Slope-Intercept Form:** Uses the classic \( y = mx + b \) format to plot the intercept on the y-axis and use the slope to find other points on the line. This method is excellent for manual graphing.
• **Table of Values:** Create a simple table by choosing x-values and calculating corresponding y-values to plot on the graph manually or verify using a calculator.
• **Intersecting Lines:** Compare the graph of your linear equation with another to find where they intersect, a useful technique for solving systems of equations.

Using these techniques helps in predicting how changes in the equation affect the graph and fosters a deeper comprehension of linear relationships.

Calculator Functions

Graphing calculators are powerful tools that simplify the process of solving and analyzing equations. Here, we'll delve into some essential functions and how they enhance your understanding:

• **Y= Menu:** This feature allows you to input and manage multiple equations simultaneously. It is the starting point for graphing.
• **Graph Button:** Essential for visualizing the equation once entered. It helps you see where lines intersect axes quickly.
• **CALC Menu / Zero Function:** Within this menu, selecting the 'zero' option facilitates finding the x-intercept efficiently. This function's precision is crucial when estimating zeros in more complex equations.

Mastering these calculator functions enables you to explore a wide range of mathematical problems more easily, providing clear insights into linear equations and beyond.