Problem 48 - College Algebra

Question

After graphing it, use the \(2^{\text {nd }}\) CALC button and 1:value button, hit ENTER. At the lower part of the screen you will see " \(\mathbf{x}=\) " and a blinking cursor. You may enter any number for \(x\) and it will display the \(y\) value for any \(x\) value you input. Use this and plug in \(x=0\), thus finding the \(y\) -intercept, for each of the following graphs. $$ Y_{1}=-2 x+5 $$

Step-by-Step Solution

Answer

The y-intercept is 5.

Step 1: Understand the Function

The function given is \( Y_1 = -2x + 5 \). This is a linear function of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For this function, the slope \( m \) is \(-2\) and the y-intercept \( b \) is \(5\).

Step 2: Use the Calculator

To find the y-intercept using a graphing calculator, input the equation \( Y_1 = -2x + 5 \) into the calculator. Open the graphing mode and plot the function on the graph screen.

Step 3: Access CALC and VALUE Features

Press the \(2^{\text{nd}}\) button followed by the CALC button on the graphing calculator. Then select the 'VALUE' option by pressing the number \(1\). This allows you to find the y-value for any specified x-value.

Step 4: Enter x-value

When prompted with \( x = \) and a blinking cursor, enter \(0\) to find the y-intercept. Press ENTER after typing \(0\).

Step 5: Interpret Calculator Output

The calculator will display the corresponding \( y \)-value. For \( x = 0 \), substitute \(0\) into the function \( Y_1 = -2x + 5 \). This gives \( Y_1 = -2(0) + 5 = 5 \). Thus, the y-intercept is \(5\).

Key Concepts

Understanding the Y-Intercept

The y-intercept is an essential concept when graphing linear functions. It tells us where the graph of a linear equation crosses the y-axis. In simpler terms, it's the point on the graph where the value of x is zero.
To locate the y-intercept of a function, you'll substitute zero for x in the equation. This makes it straightforward since the calculation only involves the constant term.
For instance, in the function given by the equation \( y = mx + b \), the y-intercept is represented by the term \( b \).
Using our example \( Y_1 = -2x + 5 \), when x is zero, the y-intercept is \( 5 \). Always remember, identifying the y-intercept helps in quickly sketching the graph of a line.

Using a Graphing Calculator

A graphing calculator is a valuable tool when dealing with linear equations. It assists you in visualizing the function and calculating specific values easily.
To use a graphing calculator to find where a graph crosses the y-axis, follow these steps:

• Enter the equation into the graphing function of the calculator
• Navigate to the graph screen to see the line plotted
• Utilize the CALC and VALUE features to explore specific points on the graph

By entering the function \( Y_1 = -2x + 5 \), you can quickly find the y-intercept by using the CALC and value features
This method provides an accurate visual representation and lets you interact with the function to gain deeper insights.

The Structure of a Linear Equation

Linear equations provide a straightforward way to represent relationships between two variables. They form straight lines when graphed.
These equations typically take the form \( y = mx + b \), where:

• \( m \) denotes the slope of the line
• \( b \) is the y-intercept

This structure is uniform, making it easier to predict and understand the behavior of a line just by looking at the equation.
Linear equations are prevalent in many real-life applications because they describe consistent relationships."
In our example equation \( Y_1 = -2x + 5 \), the coefficient \(-2\) is the slope, and the constant \(5\) is the y-intercept. Recognizing these parts helps in the quick and accurate graphing of the linear function.

Understanding the Slope

The slope of a line is a crucial factor in understanding how a line behaves. It represents the line's steepness and direction.
The slope is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.

• If the slope is positive, the line rises from left to right
• If the slope is negative, the line falls from left to right

In slope-intercept form, \( y = mx + b \), the \( m \) stands for slope.
For the linear equation \( Y_1 = -2x + 5 \), the slope is \(-2\). This tells us that for every unit increase in x, y decreases by 2 units.
Understanding the slope is essential in interpreting how changes in x affect changes in y, adding depth to the analysis of a linear function.