Problem 10 - Intermediate Algebra

Question

Graph each of the following linear and quadratic functions. $$f(x)=-2 x-4$$

Step-by-Step Solution

Answer

Graph is a line through points (0, -4) and (1, -6), crossing x-axis at (-2, 0).

Step 1: Identify the type of function

The function given is \(f(x) = -2x - 4\). This is a linear function because it can be written in the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants.

Step 2: Determine the slope and y-intercept

For the linear function \(f(x) = -2x - 4\), the slope \(m\) is -2 and the y-intercept \(b\) is -4. This means the graph will cross the y-axis at -4, and for every unit increase in \(x\), \(f(x)\) decreases by 2 units.

Step 3: Calculate two points and draw the line

To graph the function, calculate at least two points. You already know one point where it intersects the y-axis: (0, -4). For a second point, choose another value of \(x\), such as \(x = 1\). Then \(f(1) = -2(1) - 4 = -6\), giving the point (1, -6). Draw a line through these two points to extend the graph.

Step 4: Draw the x-axis intersection

To know where the line intersects the x-axis, set \(f(x) = 0\) and solve for \(x\):\[0 = -2x - 4\]\[2x = -4\]\[x = -2\]This tells us the line crosses the x-axis at (-2, 0). Add this point to the graph as well.

Step 5: Finalize the graph

Using the points (0, -4), (1, -6), and (-2, 0), draw a straight line to represent the function \(f(x) = -2x - 4\). Ensure the line extends in both directions and includes arrows to indicate it continues indefinitely.

Key Concepts

Slope

The concept of the slope is fundamental in understanding linear functions. In its essence, the slope tells us how steep a line is. More specifically, it represents the rate of change between two variables. For a linear function like \( f(x) = -2x - 4 \), the slope \( m \) is -2. This negative slope means that as you move from left to right along the graph, the function decreases. For every single unit increase in \( x \), the value of \( f(x) \) decreases by 2 units.
To visualize this:

• If the slope is positive, the line slopes upward from left to right.
• If the slope is negative, the line slopes downward from left to right.
• If the slope is zero, the line is horizontal and does not rise or fall.

The slope is a crucial parameter because it helps us predict the behavior of the line without needing to graph the entire function.

Y-Intercept

The y-intercept of a linear function is the point at which the graph of the function crosses the y-axis. For the linear function \( f(x) = -2x - 4 \), solving for the y-intercept is straightforward. Here, the y-intercept is -4.To find this:

• Set \( x = 0 \) in the function, since the y-intercept is where the line crosses the y-axis.
• Calculate \( f(0) = -2(0) - 4 \), which simplifies to \( -4 \).

This tells us that the point \( (0, -4) \) is on the graph. The y-intercept provides a starting point for graphing the line and allows us to see at which point the linear function intersects the y-axis.

X-Intercept

The x-intercept is another critical component when dealing with linear functions. It represents the point where the graph crosses the x-axis. For our function, \( f(x) = -2x - 4 \), finding the x-intercept involves setting the function equal to zero and solving for \( x \).Here's how you find it:

• Set \( f(x) = 0 \).
• Solve the equation \( 0 = -2x - 4 \) to find \( x \).
  By rearranging, you get \( 2x = -4 \), and then \( x = -2 \).

This calculation shows that the x-intercept is at the point \( (-2, 0) \). The x-intercept is useful for visualizing where on the graph the function crosses the x-axis, providing another key point to help sketch the linear function.