Problem 50 - 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. \(\mathrm{Y}_{1}=\frac{x+5}{2}\)

Step-by-Step Solution

Answer

The y-intercept of the graph is 2.5.

Step 1: Access Graphing Mode

Turn on your graphing calculator and navigate to the 'Y=' menu. This is where you will input the equation you want to graph.

Step 2: Enter the Equation

In the 'Y=' menu, input the equation \(Y_1=\frac{x+5}{2}\). Make sure to enter it correctly by using parentheses around the numerator: (x+5)/2.

Step 3: Graph the Equation

After inputting the equation, press the 'GRAPH' button. The calculator will display the graph of the equation \(Y_1=\frac{x+5}{2}\) on the screen.

Step 4: Access the Calculation Menu

Press the '2nd' button followed by the 'CALC' button (above the TRACE button on most calculators) to access the calculation menu.

Step 5: Select Value Option

From the CALC menu, select the '1:value' option by either pressing '1' or using the arrow keys to highlight it and then pressing 'ENTER'.

Step 6: Enter x=0

At the bottom of the screen, you'll see 'x=' with a blinking cursor. Enter '0' using the keypad and press 'ENTER' to calculate the corresponding y-value.

Step 7: Read the Y-Intercept

Once you press 'ENTER', the calculator will display the y-value corresponding to \(x=0\), which is the y-intercept of the graph. For \(Y_1=\frac{x+5}{2}\), this value should be \(\frac{5}{2}\) or 2.5.

Key Concepts

Graphing Calculator

A graphing calculator is a fantastic tool that can help visualize and solve mathematical problems, especially when it comes to graphing linear equations. These calculators are equipped with a screen capable of displaying graphs and functions. They have a special functionality that allows us to directly input equations and observe their graphs. With features like zoom, trace, and calculation options, you can explore different aspects of a function. This makes understanding the behavior and characteristics of linear functions much more intuitive.

• These calculators come with a graph display which can clearly show how the graph of an equation behaves.
• They allow input of various types of equations not limited to linear, including polynomial, trigonometric, and exponential functions.
• One key feature is the "CALC" menu, which lets you calculate specific values along the graph.

Using a graphing calculator enhances your learning by providing a hands-on and visual experience of equations and their respective graphs.

Y-Intercept

The y-intercept is a fundamental concept in understanding linear equations. It represents the point where a line crosses the y-axis, which means the x-coordinate of this point is always zero. Finding the y-intercept is a critical step, especially for graphing and analyzing linear functions, as it gives a starting point of the line on the graph.
For any linear equation in the form of \(y = mx + b\), the y-intercept is represented by \(b\). This value is what you would see when you set \(x=0\). For example, if you have the equation \(Y_1=\frac{x+5}{2}\), you can find the y-intercept by calculating \(Y\) when \(x=0\). This gives you \(Y = \frac{0+5}{2} = \frac{5}{2}\) or 2.5.

• The y-intercept tells you where the graph will cross the y-axis.
• In practical terms, it often represents a starting point or initial condition in real-world scenarios.
• On the graphing calculator, you can find this by using the \'value\' function, inputting \(x=0\) and reading the resultant \(y\) value.

Linear Functions

Linear functions are a type of function whose graph is a straight line. This simplicity makes them easy to work with, yet they are powerful enough to model a wide range of real-world situations. The general form of a linear function is \(y = mx + b\). In this equation:

• \(m\) represents the slope of the line, showing how steep the line is.
• \(b\) is the y-intercept, indicating where the line crosses the y-axis.

Understanding these components helps in interpreting linear functions. Linear functions have a constant rate of change, defined by the slope.
Graphically, linear functions are straightforward as they maintain a consistent slope, unlike quadratic or polynomial functions that can curve upward or downward. This predictability makes them invaluable in trends, business calculations, and more.
For the equation \(Y_1 = \frac{x+5}{2}\), when graphed, it shows a line with a set slope and a specific y-intercept. Recognizing these features aids significantly in real-world applications where linear trends are observed.

Equation Input

Equation input into a graphing calculator is straightforward but crucial for accurate graphing and calculations. Here’s how you input a linear equation like \(Y_1 = \frac{x+5}{2}\):
You start by accessing the 'Y=' menu, found on all graphing calculators. This menu is your go-to for entering any equations you want to explore. After navigating to this menu:

• Input the equation using parentheses to promote accuracy: \((x+5)/2\).
• Use the keypad for numbers, operations, and to toggle between variables.
• Once the equation is entered correctly, you proceed to visualize it by pressing the 'GRAPH' button.

This step-by-step input of linear equations allows for error checking and eventual understanding of how the input affects the output graph. If errors occur, re-checking your input can usually resolve them. Properly entering equations is fundamental, ensuring precise results, and aiding your learning of graph functions effectively.