Problem 49 - 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. $$ y=-3 x+1, y=-x+1, y=-5 x+1, y=-7 x+1 $$

Step-by-Step Solution

Answer

Steeper lines have more negative slopes; they all cross at (0, 1).

Step 1: Understand Slope and Y-intercept

The equation of a line is expressed as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope \(m\) determines the steepness and direction of the line, while \(b\) is where the line crosses the y-axis.

Step 2: Analyze the Common Element

All equations have the form \(y = mx + 1\), indicating they share the same y-intercept \(b = 1\). Therefore, all lines cross the y-axis at the point \((0, 1)\).

Step 3: Compare the Slopes

The slopes for the equations are \(-3, -1, -5,\) and \(-7\). A negative slope indicates that the line falls (or decreases) from left to right. The greater the absolute value of the slope, the steeper the line.

Step 4: Graph Each Line

Using a graphing calculator, plot each equation. Observe how each line passes through the point \((0, 1)\), but they have different steepness due to varying slopes. The line \(y = -x + 1\) is the least steep, while \(y = -7x + 1\) is the steepest.

Step 5: Draw Conclusions

Increasing the absolute value of a negative slope results in a line that becomes steeper. As \(m\) becomes more negative, the angle at which the line falls increases, leading to a steeper descent.

Key Concepts

Slope

The slope is a crucial part of a linear equation. It determines how sharply a line rises or falls on a graph. When looking at a linear equation in the form of \(y = mx + b\), \(m\) represents the slope.
The slope \(m\) tells us:

• If the line rises or falls as we move from left to right on the graph.
• A positive slope means the line will rise.
• A negative slope means the line will fall.
• The larger the absolute value of the slope, the steeper the line.

For example, in the equation \(y = -3x + 1\), the slope is \(-3\). This negative value indicates the line will fall, and the absolute value tells us how steep the fall is.

Y-intercept

The y-intercept is the point where a line crosses the y-axis. In the equation \(y = mx + b\), \(b\) represents the y-intercept.
It provides a clear starting point for drawing or understanding a graph.

• If \(b = 1\), like in our sample equations, the line crosses the y-axis at \((0, 1)\).
• The y-intercept remains the same for all equations with the same value of \(b\).

Understanding the y-intercept helps to quickly plot the line on a graph by providing a fixed point, no matter the slope.

Graphing Calculator

A graphing calculator is a powerful tool that assists in visualizing equations, especially linear ones. It helps plot lines and observe their behavior in real-time.
By entering equations into a graphing calculator like \(y = mx + 1\), you can:

• Instantly see how the slope changes affect line steepness.
• Observe the same y-intercept across different lines.
• Compare different equations visually and understand concepts better.

Using a graphing calculator simplifies understanding complex relationships and trends by turning equations into visible graphs.

Line Steepness

Line steepness is directly impacted by the slope in a linear equation. The steeper a line, the greater its absolute slope value.
Consider this:

• The equation \(y = -x + 1\) has a slope of \(-1\) and is less steep compared to \(y = -7x + 1\) which has a slope of \(-7\).
• A greater absolute value of slope results in a steeper line on the graph.
• Steepness also determines the rate of change; a steeper line indicates a faster change in y regarding changes in x.

Visualizing different slopes helps clarify how line steepness varies in linear equations and impacts graph interpretation.