Problem 11 - Elementary and Intermediate Algebra

Question

Consider the linear function \(f(x)=-6 x-4\). a. What is the \(y\) -intercept of its graph? b. What is the \(x\) -intercept of its graph?

Step-by-Step Solution

Answer

The y-intercept is -4, the x-intercept is \(-\frac{2}{3}\).

Step 1: Identify the y-intercept

The y-intercept of a linear function is the point where the graph intersects the y-axis. For a function in the form of \( f(x) = mx + b \), the y-intercept is \( b \). In this function, \( f(x) = -6x - 4 \), so the y-intercept is \( -4 \). The coordinates for the y-intercept are \((0, -4)\).

Step 2: Find the x-intercept by setting f(x) to zero

To find the x-intercept, set the function equal to zero: \( f(x) = -6x - 4 = 0 \). This will give you the x-coordinate where the graph intersects the x-axis.

Step 3: Solve for x

Rearrange the equation \(-6x - 4 = 0\) to solve for x. Add 4 to both sides: \(-6x = 4\). Then, divide both sides by -6: \( x = -\frac{4}{6} \). Simplify to \( x = -\frac{2}{3} \). The coordinates for the x-intercept are \((-\frac{2}{3}, 0)\).

Key Concepts

Understanding the y-intercept

The y-intercept is a fundamental concept in linear functions. It's the point where the graph of a linear equation crosses the y-axis. This happens when the input value, or x-value, is zero. In simpler terms, if you substitute 0 for x in your linear function, you’ll arrive at the y-intercept.

For any linear equation of the standard form \(f(x) = mx + b\), the y-intercept is represented by \(b\). This is because the x-term cancels out, leaving us with just \(b\).

• For instance, in the equation \(f(x) = -6x - 4\), setting \(x = 0\) gives a result of \(f(0) = -4\).
• This means the y-intercept here is -4.

Keep in mind, the coordinates of the y-intercept are always expressed as \((0, b)\). In the current example, the coordinates are \((0, -4)\).

Knowing the y-intercept enables you to start plotting the linear graph more easily.

Exploring the x-intercept

The x-intercept is another crucial aspect of understanding linear functions. This is the point where the graph intersects the x-axis. Here, the y-value is zero. To find the x-intercept, you need to set the function equal to zero and solve for x, essentially determining when the output is zero.

Let's consider the equation \(f(x) = -6x - 4\). Set this equation equal to zero to find the x-intercept:

• Start with \(-6x - 4 = 0\).
• Add 4 to each side: \(-6x = 4\).
• Divide each side by -6 to isolate x: \(x = -\frac{4}{6}\).
• Simplify to obtain \(x = -\frac{2}{3}\).

Therefore, the x-intercept is \(x = -\frac{2}{3}\), and its coordinates are \((-\frac{2}{3}, 0)\).

Understanding x-intercepts allows you to grasp where the graph crosses the x-axis, which is directly connected to understanding the roots or solutions of the equation.

Graphing Linear Equations Made Easy

Graphing linear equations involves plotting points on a coordinate grid to visualize the equation. Start with the y-intercept, as it's a straightforward point that you already know. Then, use other known values, such as the x-intercept and additional points from the equation, to form a straight line.

Let's break it down for \(f(x) = -6x - 4\):

• Begin by plotting the y-intercept \((0, -4)\) on the graph. This is your starting point.
• Next, plot the x-intercept \((-\frac{2}{3}, 0)\), which gives you a second point.
• Remember, a linear function like \(f(x) = -6x - 4\) creates a straight line. So, you only need these two points to draw the line precisely.
• If you’d like, pick another x-value, substitute it into the equation, and find a third point for more accuracy.

Graphing will provide a visual representation of the equation, showing how the line slopes and crosses the axes. This process helps in understanding the relationship between x and y within the equation.