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

Step-by-Step Solution

Answer

The y-intercept is 2.5.

Step 1: Graphing the Equation

First, you need to graph the equation \(Y_1 = \frac{x+5}{2}\) on your graphing calculator. Enter the function into the calculator by accessing the Y= menu and typing \((x + 5) / 2\) into \(Y_1\).

Step 2: Access the CALC Menu

Press the \(2^{\text{nd}}\) button followed by the TRACE (CALC) button to access the CALC menu. This will allow you to use various calculator functionalities related to the graph.

Step 3: Use the '1: value' Feature

Select '1:value' from the CALC menu. This feature allows you to input a specific \(x\) value and find the corresponding \(y\) value on the graph.

Step 4: Input x=0

After selecting '1:value', type \(x = 0\) when prompted at the bottom of the screen (where it shows \(x=\) with a blinking cursor). Input this value and press ENTER.

Step 5: Find the y-intercept

The calculator will display the \(y\) value that corresponds to \(x = 0\). This \(y\) value is the \(y\)-intercept of the graph. For the function \(Y_1 = \frac{x+5}{2}\), substituting \(x = 0\) gives \(y = \frac{0+5}{2} = \frac{5}{2} = 2.5\).

Key Concepts

y-intercept

In algebra, the y-intercept of a graph is a crucial point. It is where the graph crosses the y-axis. To find the y-intercept, you set the value of \(x\) to 0 in the equation of the line or curve. This gives you the \(y\) coordinate of the intercept.

In our example with the equation \(Y_1 = \frac{x+5}{2}\), we substitute \(x = 0\). When we calculate, it becomes \(y = \frac{0+5}{2} = 2.5\). Therefore, the y-intercept here is \(2.5\). This means the graph will cross the y-axis at the point \((0, 2.5)\).

Finding the y-intercept helps in understanding how the graph is positioned on the coordinate axis. It also gives an initial value which can be useful for sketching and interpreting the graph formation.

value feature

The 'value' feature on graphing calculators is a handy tool. It allows you to find precise y-values by plugging in specific x-values. This functionality is accessed through the CALC menu.

Here's a quick guide on how you can use this feature for your graphing calculators:

• First, ensure you have graphed your function. This connects you to the correct visual representation.
• Navigate to the CALC menu via the \(2^\text{nd}\) button and then "TRACE".
• Select "1:value" to enter your desired \(x\) value.
• The calculator will then calculate and show the corresponding \(y\) value for the inputted \(x\) on the graph.

This feature is excellent because it does instant calculations that would otherwise require manual computations. It also aids in verifying y-coordinates effectively.

graph equations

Graphing equations is a fundamental part of algebra, as it helps visualize mathematical relationships. When graphing an equation, you essentially draw its graphical representation on the coordinate plane using a graphing calculator.

For the equation \(Y_1 = \frac{x+5}{2}\), graphing involves plotting a series of points that satisfy the equation and connecting them to form a line.

• Start by entering the equation into the calculator's function menu, which is usually labeled "Y=".
• Ensure that the function is correctly typed, in this case, "\((x+5)/2\)".
• After inputting, press the 'GRAPH' button to display the equation graphically on the screen.

The line will provide a visual interpretation of how the equation behaves. It's a way to immediately see trends and relationships between x and y-values for the equation.

calculators in algebra

Graphing calculators are indispensable tools in algebra. They allow for dynamic interaction with equations and functions. Their value is seen in bridging abstract algebraic ideas with visual understanding.

Using calculators in algebra provides several benefits:

• They handle complex calculations, reducing human error in manual computations.
• Graphical visualization helps in understanding function behaviors, such as slopes, intercepts, and intersections.
• Interactive features like zoom in/out or tracing enable exploration of different graph areas, enhancing comprehension.

Moreover, calculators offer an efficient way to solve algebraic problems quickly, which saves time during homework or exams. Building familiarity with calculators can significantly improve problem-solving skills and confidence in tackling algebraic concepts.