Problem 20 - College Algebra with Modeling and Visualization

Question

Exercises \(19-32:\) Graph the linear function by hand. Identify the slope and y-intercept. $$ f(x)=-\frac{3}{2} x $$

Step-by-Step Solution

Answer

The slope is \(-\frac{3}{2}\) and the y-intercept is 0.

Step 1: Understand the Function Format

The given function is in the form of a linear equation, specifically the form of a line which is \( f(x) = mx + b \). Here, \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Identify Slope and Y-intercept

In the function \( f(x) = -\frac{3}{2}x \), \( m = -\frac{3}{2} \) and \( b = 0 \). This means the slope of the line is \(-\frac{3}{2}\) and the y-intercept is at the origin (0,0).

Step 3: Plot the Y-intercept

Since the y-intercept is 0, plot the point (0,0) on the graph. This is where the line will intersect the y-axis.

Step 4: Use the Slope to Find Another Point

The slope \(-\frac{3}{2}\) means that for a run of 2 units to the right, the rise will be -3 units. From the point (0,0), move 2 units to the right (x = 2) and then move 3 units down (y = -3), plotting the point (2, -3).

Step 5: Draw the Line

Connect the points (0,0) and (2,-3) with a straight line. Extend this line in both directions, this is the graph of the function \( f(x) = -\frac{3}{2}x \).

Step 6: Verify the Line's Correctness

Check that additional points line up with the slope of \(-\frac{3}{2}\). For instance, from (2,-3) moving over by 2 results in (4,-6). Verify that this new point also lies on the drawn line.

Key Concepts

Understand the Slope in Linear Functions

The slope is one of the key ingredients of a linear equation, expressed as the letter \( m \) in the formula \( f(x) = mx + b \). This little number describes how slanty or steep a line is on a graph. You can think of it like the steepness of a hill when you're biking.

• The slope tells us how many units a line moves up or down for every unit you move to the right. It's also known as "rise over run."
• A positive slope means the line is climbing upward as you go to the right. A negative slope means it's going downhill.
• In our function \( f(x) = -\frac{3}{2}x \), the slope \( m \) is \(-\frac{3}{2}\). This negative value means that for every 2 units you move to the right, you move 3 units down.

Understanding slope really helps us predict and draw where the line will travel on a graph.

Understanding the Y-intercept

The y-intercept is where our line on the graph crosses the y-axis, that's the vertical line running up and down. It is found at the point \( (0, b) \) where \( b \) is the y-intercept value in the equation \( f(x) = mx + b \).

• This point is super important because it's like the starting place of your line when you're beginning to graph.
• For our example function \( f(x) = -\frac{3}{2}x \), the y-intercept \( b \) is 0. This places the intercept at the origin (0,0), which is right where the x-axis and y-axis meet.

Sometimes the y-intercept can be confusing if it isn't so neatly at zero, but once you know where to put your first dot, it's easier to continue graphing with the slope.

Graphing Linear Equations Made Simple

When you graph a linear equation, what you're really doing is turning math into a visual picture. Take the pieces, like the slope and y-intercept, and these help plot the line step by step.

• Start at the y-intercept. In our case, that's (0,0). Put a point there since it's your starting mark.
• Utilize your slope \( m = -\frac{3}{2} \). From (0,0), move 2 units to the right, and because our slope is negative, go 3 units down to find another point like (2,-3).
• Draw a line through these points and keep it going in both directions. The line illustrates all points that satisfy the equation.
• To make sure your work is accurate, check by picking another x-value, calculating the y with the equation, and verifying if it fits on the line.

Graphing linear equations can seem like magic at first, but remember, each line simply follows a story told by the slope and y-intercept. Every graph is a different story about how x and y are related.