In Exercises 31 and 32, the line contains the origin and the...

In Exercises 31 and 32, the line contains the origin and the point in the upper right corner of the grapher screen. Which line shown here has the greater slope? Explain. $$\begin{array}{l}{[-20,20] \text { by }[-35,35\rfloor \qquad[-5,5] \text { by }[-20,20]} \\ \quad\quad\quad\quad{\text { (a) }}\quad\quad\quad\quad\quad\quad\quad\quad {\text { (b) }}\end{array} $$

In Exercises 31 and 32, the line contains the origin and the point in the upper right corner of the grapher screen.
Which line shown here has the greater slope? Explain.
$$\begin{array}{l}{[-20,20] \text { by }[-35,35\rfloor \qquad[-5,5] \text { by }[-20,20]} \\ \quad\quad\quad\quad{\text { (a) }}\quad\quad\quad\quad\quad\quad\quad\quad {\text { (b) }}\end{array} $$

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Key Concepts

Slope

Slope is a measure of the steepness or inclination of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. It helps in determining how rapidly a line increases or decreases and is a central concept in coordinate geometry and linear functions.

Line Through Two Points

When a line is defined by two points, its slope can be found by computing the difference in the y-coordinates divided by the difference in the x-coordinates. This fundamental idea is used to analyze and compare the steepness of different lines, regardless of how the graph is scaled.

Graph Scaling and Coordinate Systems

Graph scaling involves setting different ranges or intervals on the x-axis and y-axis, which can affect the visual appearance of a line’s steepness. Understanding that the slope calculation is independent of the scaling of the graph is important; the actual numerical slope remains the same regardless of how the coordinate system is set up.

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