Problem 47 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(g(x)=0.5 x-3\)

Step-by-Step Solution

Answer

The x-intercept is (6, 0) and the y-intercept is (0, -3).

Step 1: Find the y-intercept

To find the y-intercept of the equation \( g(x) = 0.5x - 3 \), set \( x = 0 \) because the y-intercept is where the graph crosses the y-axis.\[ g(0) = 0.5(0) - 3 = -3 \] Thus, the y-intercept is the point \((0, -3)\).

Step 2: Find the x-intercept

To find the x-intercept, set \( g(x) = 0 \) and solve for \( x \) because the x-intercept is where the graph crosses the x-axis.\[ 0 = 0.5x - 3 \]Add 3 to both sides:\[ 3 = 0.5x \]Divide both sides by 0.5:\[ x = \frac{3}{0.5} = 6 \]Thus, the x-intercept is the point \((6, 0)\).

Step 3: Graph the Equation

To graph \( g(x) = 0.5x - 3 \), plot the intercepts found in Steps 1 and 2: the y-intercept \((0, -3)\) and the x-intercept \((6, 0)\). Draw a straight line through these points. The line represents the graph of the equation with a slope of 0.5.

Key Concepts

Understanding Intercepts

When working with linear equations, intercepts are key points where the graph crosses the axes. They are critical for sketching graphs and understanding the geometry of the line. Let's break down the two types of intercepts:

• Y-Intercept: This is the point where the line crosses the y-axis. You find it by setting \( x = 0 \) in the equation. For the equation \( g(x) = 0.5x - 3 \), substituting zero for \( x \) simplifies the equation to \( g(0) = -3 \). Thus, the y-intercept is \((0, -3)\).
• X-Intercept: This is where the line crosses the x-axis. To find this, set the equation \( g(x) = 0 \). Solving for \( x \) involves basic algebra: \( 0 = 0.5x - 3 \), leading to \( x = 6 \). So the x-intercept is \((6, 0)\).

Both intercepts provide starting points for graphing, helping you understand the graph's orientation and position.

Plotting Through Graphing

Graphing linear equations can be incredibly easy once you have your intercepts. These are your blueprint coordinates on the graph that allow you to build a clear picture of the line.To graph \( g(x) = 0.5x - 3 \), you'll simply plot your calculated intercepts:

• The y-intercept at \((0, -3)\)
• The x-intercept at \((6, 0)\)

Connect these points with a straight edge to form a line. You'll see that the line stretches infinitely while running through these two anchor points.Graphing in this way helps in visualizing the linear relationship defined by the equation. It makes abstract numbers tangible and enhances understanding of real-world linear relationships.

Deciphering the Slope

The slope of a line in a linear equation is a measure of its steepness and direction. For the function \( g(x) = 0.5x - 3 \), the slope is indicated by the number before the \( x \), which is 0.5.The slope can be thought of as:

• Rise over Run: It is the ratio of the vertical change to the horizontal change between two points on the line. Here, a slope of 0.5 means you "rise" 0.5 units vertically for every 1 unit you "run" horizontally.
• Positive Direction: Since 0.5 is positive, the line will slant upwards from left to right.

Understanding slope is fundamental because it describes how the change in one variable affects the other in a linear manner. It's essential for predicting future points on the graph and for insight into how rapidly values increase or decrease with changes in \( x \).