Problem 43 - Algebra 1

Question

Use a table of values to graph the equation. \(y=-\frac{3}{4} x+1\)

Step-by-Step Solution

Answer

The graph of the equation \(y = -\frac{3}{4}x + 1\) is a straight line that slopes downward from left to right, intersecting the y-axis at y=1. This was confirmed after plotting the points (-2, 2.5), (0, 1), and (2, -0.5) obtained from our table of values.

Step 1: Identify the slope and y-intercept

The equation provided is \(y = -\frac{3}{4}x + 1\). Here, the number in front of x (-3/4) is the slope (m), and the constant at the end of the equation (1) is the y-intercept (b). Thus, for every 4 units that x increases, y will decrease by 3 units, and when x=0, y=1.

Step 2: Construct a table of values

Choose a range of x values. For example, [-2, 0, 2]. Now substitute each x value into the equation to find the corresponding y value. For x=-2, substituting into the equation gives \(y = -\frac{3}{4}*(-2) + 1 = 2.5\). For x=0, \(y = -\frac{3}{4}*0 + 1 = 1\). For x=2, substituting gives \(y = -\frac{3}{4}*2 + 1 = -0.5\). Our table of values is thus as follows: (-2, 2.5), (0, 1), (2, -0.5).

Step 3: Plot the points on a graph

The coordinate pairs from the table of values become points on the graph. Plot these points and draw a straight line through them. The line represents the equation. When you've plotted points (-2, 2.5), (0, 1), and (2, -0.5), a downward sloping straight line will emerge, which is accurate given the negative slope (-3/4) of the equation.

Key Concepts

Understanding the Slope-Intercept Form

The slope-intercept form is a special format for a linear equation of the form \(y = mx + b\). It's extremely useful because it directly provides two important pieces of information:

• The slope \(m\), which tells us how steep the line is.
• The y-intercept \(b\), which is the point where the line crosses the y-axis.

For the equation \(y = -\frac{3}{4}x + 1\), the slope \(m = -\frac{3}{4}\). This means that for every 4 units increase in \(x\), \(y\) decreases by 3 units. The negative sign indicates the line slopes downward. The y-intercept \(b = 1\) signifies that the line will cross the y-axis at (0, 1). This characteristic of the equation provides a clear starting point for graphing.

Creating a Table of Values

To graph the equation \(y = -\frac{3}{4}x + 1\), constructing a table of values is a practical step. This involves selecting a few values for \(x\), and solving for their corresponding \(y\):

• Choose some \(x\) values. Here we chose \([-2, 0, 2]\) for simplicity.
• Substitute each \(x\) into the equation to calculate \(y\).
• For \(x = -2\), \(y = 2.5\); for \(x = 0\), \(y = 1\); and for \(x = 2\), \(y = -0.5\).

Plotting works best when you have at least 3 points to ensure accuracy. The table of values highlights these coordinate pairs: (-2, 2.5), (0, 1), and (2, -0.5), providing the necessary coordinates to plot your line.

Plotting on the Coordinate Plane

After generating a table of values, the next step is to plot these points on a coordinate plane. Here's how to do it:

• Draw two perpendicular lines intersecting at the origin: one horizontal (x-axis) and one vertical (y-axis).
• Locate each pair from your table of values: (-2, 2.5), (0, 1), and (2, -0.5).
• Mark these points on your graph. It's important that the scale is consistent on both axes to correctly represent the points.

Once you've marked the points, use a ruler to draw a straight line connecting them. This line represents the equation of the graph. Check that the line has the correct slope: from one point, the line should drop 3 units vertically for every 4 units it extends horizontally. The plotted line shows the relationship defined by your original equation, giving a visual representation of the slope-intercept concept.