Problem 42 - College Algebra

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of each equation. $$ g(x)=2 x+4 $$

Step-by-Step Solution

Answer

X-intercept is (-2, 0) and Y-intercept is (0, 4).

Step 1: Understanding Intercepts

To find the intercepts of the given equation, we need to understand what the intercepts are. The x-intercept is the point where the graph of the equation intersects the x-axis, which means y equals zero. The y-intercept is the point where the graph intersects the y-axis, which means x equals zero.

Step 2: Finding the X-Intercept

To find the x-intercept, set the output of the function, g(x), which is equals to 0. So, solve the equation: \[ 0 = 2x + 4 \] Subtract 4 from both sides:\[ 0 - 4 = 2x + 4 - 4 \] This simplifies to: \[ -4 = 2x \] Next, divide both sides by 2:\[ x = \frac{-4}{2} \] \[ x = -2 \] The x-intercept is (-2, 0).

Step 3: Finding the Y-Intercept

To find the y-intercept, set x equal to zero in the function g(x): \[ g(0) = 2(0) + 4 \] Simplify the expression:\[ g(0) = 0 + 4 \] \[ g(0) = 4 \]So, the y-intercept is (0, 4).

Key Concepts

x-intercept

The concept of the x-intercept is quite fascinating as it deals with where a graph touches the x-axis. Understanding x-intercepts involves identifying the point at which the value of the function equals zero. This is where the linear equation crosses the x-axis.
To determine the x-intercept of a linear equation such as \( g(x) = 2x + 4 \), set the output, \( g(x) \), to zero. The equation then becomes \( 0 = 2x + 4 \). You need to

• subtract 4 from both sides to isolate the term with \( x \), leading to \( -4 = 2x \),
• then divide by 2 to solve for \( x \), giving you \( x = -2 \).

The resulting x-intercept is \((-2, 0)\), indicating that the graph crosses the x-axis at the point where \( x = -2 \).
This point is crucial because it visually represents the solution to the equation when \( g(x) \) is zero.

y-intercept

The y-intercept is another key component in understanding the graph of a linear equation. It signifies the point where the graph meets the y-axis. At this point, the value of \( x \) is always zero, making it easier to calculate.
For the equation \( g(x) = 2x + 4 \), finding the y-intercept is straightforward. Simply set \( x \) to zero and solve for \( g(x) \). This means you calculate \( g(0) = 2(0) + 4 \).

• First, multiply \( 2 \) by \( 0 \), which is zero.
• Then, add 4 to this result, giving you \( g(0) = 4 \).

Thus, the y-intercept is \((0, 4)\). This intersection point highlights where the line crosses the y-axis, providing insight into the behavior of the function when \( x = 0 \).

linear equations

Linear equations like \( g(x) = 2x + 4 \) represent straight lines when graphed on a coordinate plane. Understanding these equations involves recognizing their consistent rate of change, which is known as the slope.
In a linear equation format \( y = mx + b \),

• \( m \) represents the slope of the line, indicating how steep the line is.
• \( b \) denotes the y-intercept, showing where the line crosses the y-axis.

For \( g(x) = 2x + 4 \), the equation tells us that the slope \( m \) is 2, meaning for every unit increase in \( x \), \( y \) will increase by 2.
Linear equations are fundamental in algebra as they model relationships with a constant rate of change. They form the basis for understanding more complex functions.
This understanding of linear equations, and how to find intercepts, aids in deciphering the relationships depicted in graphs.