Problem 49 - 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 1:value button, hit enter. At the lower part of the screen you will see " \(x=\) " 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. \(\mathrm{Y}_{1}=\frac{3 x-8}{4}\)

Step-by-Step Solution

Answer

The \(y\)-intercept is -2.

Step 1: Set Up the Equation

Identify the given equation that you need to input into the graphing calculator. The equation given is \(\mathrm{Y}_{1}=\frac{3x-8}{4}\).

Step 2: Input Equation into Calculator

Turn on your graphing calculator. Access the \(\mathrm{Y}=\) graph menu and input the equation \(\mathrm{Y}_{1}=\frac{3x-8}{4}\) into one of the slots.

Step 3: Graph the Equation

After entering the equation into the calculator's function, press the 'GRAPH' button to display the graph of the equation on the screen.

Step 4: Access Calculation Menu

Press the '2nd' button followed by the 'CALC' button to access the calculation menu on the calculator.

Step 5: Select the 'Value' Option

Press '1' to select the 'Value' option from the calculation menu. This option allows you to find the \(y\)-value for any given \(x\)-value.

Step 6: Enter \(x=0\)

With the calculation menu active, you will see \(x=\) with a blinking cursor. Enter '0' and press 'ENTER' to find the \(y\)-value when \(x=0\).

Step 7: Read the \(y\)-intercept

The calculator will display the corresponding \(y\)-value. For \(x=0\), this value is the \(y\)-intercept of the graph. Calculate \(y=\frac{3(0)-8}{4} = \frac{-8}{4} = -2\). The \(y\)-intercept is -2.

Key Concepts

Linear Equation

A linear equation is a mathematical expression that, when graphed, creates a straight line. These equations are typically in the form of \( y = mx + b \). Here, \( m \) represents the slope, or the steepness of the line, and \( b \) denotes the y-intercept, which is where the line crosses the y-axis.

Linear equations are foundational in algebra because they show a constant rate of change; as one variable increases or decreases, the other does so proportionally.

For example, the equation \( Y_1 = \frac{3x-8}{4} \) is linear because it simplifies to \( y = \frac{3}{4}x - 2 \). The term \( \frac{3}{4} \) serves as the slope, and \( -2 \) is the y-intercept. Such lines always graph as straight lines, allowing for straightforward predictions just by plugging in different values of \( x \).

Y-Intercept

The y-intercept is a key concept in graphing linear equations. It refers to the point where the line crosses the y-axis.

This occurs when \( x = 0 \). In the equation \( y = \frac{3}{4}x - 2 \), if we replace \( x \) with 0, we can easily find the y-intercept:

• Substitute \( x = 0 \) into the equation: \( y = \frac{3}{4}(0) - 2 \).
• Simplify the expression: \( y = 0 - 2 \).
• Thus, the y-intercept is \( y = -2 \).

This means the line crosses the y-axis at the point (0, -2). Understanding this point is essential as it offers insight into where a line begins when visualizing it on a graph. It's the starting point in time-related graphs, monetary predictions, and many other real-life scenarios.

Calculation Menu

The calculation menu on a graphing calculator is a powerful tool that aids in probing deeper insights from graphically represented data. When you graph a function, like a linear equation, this menu assists you in analyzing and interrogating the graph.

To access the menu, press the '2nd' button followed by the 'CALC' button. This will open the calculation menu where you can use various options to explore the graph's attributes, such as locating intersections, finding minima and maxima, and as in our exercise, determining specific values.

By selecting the 'Value' option by pressing '1', you can pinpoint precise y-values for given x-values, enabling you to explore specific points on the line. This is particularly helpful for confirming calculations, like finding the y-intercept, or observing changes over an interval.

Graphing Functions

Graphing functions is a fundamental technique in both mathematics and many applied fields, like physics and finance. It visually represents the relationship between variables.

When graphing a linear equation, you start by entering it into a graphing calculator's function menu. Following this, pressing the 'GRAPH' button will chart the equation as a straight line on the calculator's display. This visual helps comprehend how the variables relate and further aids in understanding the equation's characteristics.

• A graph shows slopes, demonstrating the rate of change.
• It illustrates y-intercepts, showing where the line hits the y-axis.
• You can interpret real-world problems through basic graph properties.

Graphing is helpful because it turns abstract algebraic concepts into visible patterns, offering clarity in analyzing and predicting outcomes from the equations.