Problem 15 - Precalculus with Limits

Question

In Exercises 15-18, complete the table. Use the resulting solution points to sketch the graph of the equation. \( y = -2x + 5 \)

Step-by-Step Solution

Answer

The solution points of the equation \( y = -2x + 5 \) are (-2,9), (0,5), and (2,1). By plotting these points and drawing a straight line through them, a sketch of the function \( y = -2x + 5 \) is obtained.

Step 1: Identify the slope and y-intercept

The equation is in the form \( y = mx + c \), where 'm' is the slope and 'c' is the y-intercept. In the equation \( y = -2x + 5 \), the slope 'm' is -2 and the y-intercept 'c' is 5.

Step 2: Choose values for x

Choose a few values for 'x'. The selection can be arbitrary, but often it's easiest to choose small numbers, including negatives, zero, and positives. Let's choose -2, 0, and 2.

Step 3: Solve for corresponding y values

Use the chosen x-values and the equation \( y = -2x + 5 \) to solve for the corresponding y-values. For x = -2, y = -2(-2) + 5 = 9. For x = 0, y = -2(0) + 5 = 5. For x = 2, y = -2(2) + 5 = 1. So, the solution points are (-2,9), (0,5), and (2,1).

Step 4: Sketch the graph

Plot the points on a two-dimensional coordinate system and draw a straight line through them. This line represents the function \( y = -2x + 5 \). Ensure the line crosses the y-axis at the y-intercept point which is 5, validating the graph.

Key Concepts

Understanding the Slope

The slope of a line in a linear equation helps us understand how the line tilts in a coordinate system. It is represented by the letter 'm' in the equation of a line, usually written as \( y = mx + c \). In our given example, \( y = -2x + 5 \), the slope \( m \) is -2.
The numerical value of the slope indicates how steep the line is:

• A positive slope means the line rises as you move to the right.
• A negative slope indicates the line falls as you move to the right.

The magnitude of the slope indicates steepness:

• A larger absolute value means a steeper line.
• When \( m = 0 \), the line is horizontal, displaying no tilt.

In this case, since \( m = -2 \), the line decreases by 2 units vertically for every 1 unit it moves horizontally to the right.

Defining the Y-Intercept

The y-intercept is where the line crosses the y-axis on a graph. In our equation \( y = -2x + 5 \), the y-intercept is 5, represented by 'c'.
The y-intercept tells us the starting point of the line on the y-axis when \( x \) is zero. This means if you plug \( x = 0 \) into the equation, you will find \( y = 5 \).
Understanding the y-intercept:

• It is the point (0, c) on the graph.
• The y-intercept is essential for sketching the graph accurately.
• It tells us where the line starts rising or falling along the y-axis.

In our graph, this information helps us locate and plot the point where the line enters the y-axis.

Finding Solution Points

Solution points are specific coordinates that lie on the line of a linear equation. For the equation \( y = -2x + 5 \), we can find these points by plugging in different values for \( x \) and solving for \( y \).
Let's use the calculated values:

• For \( x = -2 \), \( y = -2(-2) + 5 = 9 \) resulting in the point (-2, 9).
• For \( x = 0 \), \( y = -2(0) + 5 = 5 \) resulting in the point (0, 5).
• For \( x = 2 \), \( y = -2(2) + 5 = 1 \) resulting in the point (2, 1).

These solution points are crucial for drawing an accurate graph. By connecting these points, you create a complete representation of the equation's line.

Plotting in the Coordinate System

The coordinate system is a two-dimensional plane where we can visualize linear equations as lines. It is composed of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis).
For graphing our equation \( y = -2x + 5 \):

• Start by marking the y-intercept at (0, 5).
• Next, plot the solution points: (-2, 9), (0, 5), and (2, 1).
• Draw a straight line through these points.

The line should cross the y-axis at the y-intercept and confirm the slope's influence by its steeper descending direction as you move to the right.
This graph visually represents the relationship between x and y as described by the equation.