Problem 48 - Algebra 2

Question

For Exercises 48 and \(49,\) use a graphing calculator to investigate the graphs of each set of equations. Explain how changing the slope affects the graph of the line. $$ y=2 x+3, y=4 x+3, y=8 x+3, y=x+3 $$

Step-by-Step Solution

Answer

Increasing slope makes the line steeper; all share y-intercept 3.

Step 1: Understand the Slope-Intercept Form

Each equation is given in the slope-intercept form, which is \( y = mx + b \). Here, 'm' represents the slope of the line, and 'b' is the y-intercept. All four equations have the same y-intercept of 3, meaning they cross the y-axis at the point (0,3).

Step 2: Graph the First Equation

Graph the equation \( y = 2x + 3 \). The slope is 2, which means for every unit you move to the right (positive x-direction), the line moves up 2 units. The line rises fairly steeply due to the positive slope of 2.

Step 3: Graph the Second Equation

Graph the equation \( y = 4x + 3 \). The slope is 4, indicating a steeper line than the first equation. The line rises even more sharply with a positive 4-unit change in y for every 1-unit change in x.

Step 4: Graph the Third Equation

Graph the equation \( y = 8x + 3 \). The slope is 8, making the line even steeper than the previous ones. Here, the line exhibits a significant increase, going up by 8 units for each unit it moves to the right.

Step 5: Compare to a Shallower Slope

Graph the equation \( y = x + 3 \). The slope is 1, the smallest among these examples. This means the line's steepness is the least, rising by one unit on the y-axis for every unit on the x-axis.

Step 6: Analyze the Effect of Changing Slope

By comparing all graphs, you can see that increasing the slope value results in a steeper ascent, while a smaller slope results in a less steep line. All lines cross the y-axis at the same point, but they vary substantially in steepness.

Key Concepts

Graphing Linear Equations

Graphing linear equations helps us visualize relationships between variables. These equations are often written in the slope-intercept form, which is \( y = mx + b \). The beauty of this form is its simplicity: it lets us quickly identify the slope \( m \) and the y-intercept \( b \). Let's look at how to graph an equation like this.When plotting, start with the y-intercept, where the line crosses the y-axis. For example, the equations given, all intercept the y-axis at \( (0, 3) \). From there, use the slope to determine the direction and steepness of the line. For instance, a slope of 2 would mean that for every 1 unit you move to the right, the line goes up 2 units. Imagine using a graphing calculator to plot these lines; it would paint a clear picture of these relationships on a graph. This visual method is incredibly useful in both math and real-world applications.

Effect of Slope on Graph

The slope in a linear equation determines how steep or flat the line on the graph will be. It's denoted by \( m \) in the equation \( y = mx + b \). A larger slope means a steeper line, while a smaller slope results in a flatter line.Consider these examples:

• \( y = 2x + 3 \) – The slope is 2, which means the line rises steeply.
• \( y = 4x + 3 \) – With a slope of 4, this line is even steeper.
• \( y = 8x + 3 \) – This line is the steepest with a slope of 8.
• \( y = x + 3 \) – Here, the slope is 1, making it the flattest among the examples.

Each increase in slope makes the line steeper. Imagine adjusting the angle of a ramp; the greater the slope, the steeper it becomes. The same concept applies here. A steeper line represents a larger change in y relative to x.

Y-intercept in Linear Equations

The y-intercept is a key component in the slope-intercept form equation \( y = mx + b \). It's represented by \( b \) and indicates where the line crosses the y-axis. It provides us with a starting point for graphing.For all the example equations \( y = 2x + 3, y = 4x + 3, y = 8x + 3, \) and \( y = x + 3 \), the y-intercept is 3. On a graph, this means all lines cross the y-axis at the point \((0, 3)\). Recognizing this detail helps when sketching graphs by hand or verifying results on a calculator.Understanding the y-intercept's role clarifies not only the initial position of lines on a graph but also how they align with each other. If two lines share the same y-intercept, they will cross the y-axis at the same point, though their paths may diverge based on their slopes.