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Chapter 7: Problem 55

What will happen to the slope of line \(p\) if the line is shifted so that the \(y\) -intercept stays the same and the \(x\) -intercept increases? (GRAPH CAN'T COPY). F The slope will change from negative to positive. G The slope will become zero. H The slope will decrease. J The slope will increase.

Short Answer

Expert verified
The slope will decrease.

Step by step solution

01

Understanding the Problem

The problem provides a line, labeled as line \(p\). We're asked to analyze what happens to the slope of this line if it is shifted in such a way that its \(y\)-intercept remains the same but its \(x\)-intercept increases.
02

Recalling the Slope Formula

The slope \(m\) of a line, given by the equation \(y = mx + b\), is the ratio of the change in \(y\) to the change in \(x\). It can also be expressed by the formula \(-\frac{b}{a}\) when the line is in the form \(ax + by = c\). In the standard slope-intercept form, it's \(m\).
03

Effect of Increase in X-Intercept

If the \(x\)-intercept increases, this means the point where the line crosses the \(x\)-axis shifts to the right, taking more steps along the \(x\)-axis while keeping the same height (value of \(b\)).
04

Analyzing the Slope Change

As the \(x\)-intercept increases and the \(y\)-intercept stays the same, the angle of the line compared to the \(x\)-axis becomes shallower. Thus, if the line initially had a negative slope, it now becomes less negative; hence, the magnitude of the slope reduces since the line becomes closer to being horizontal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Line Intercepts
In geometry, intercepts are the points where a line crosses the axes on a coordinate plane. Specifically, the **y-intercept** is where the line meets the y-axis, while the **x-intercept** is where it touches the x-axis. These points can drastically change a line's behavior and appearance.
  • Y-Intercept: This remains constant in the given problem, meaning the point where the line crosses the y-axis doesn't shift. We signify this point in equations with the letter \(b\) from the slope-intercept form \(y = mx + b\).
  • X-Intercept: If this increases, as it does in our problem, it indicates that the line takes longer to cross the x-axis, effectively stretching along the x-direction.
Understanding how these intercepts are defined helps us visualize changes in a line's slope as one or both intercepts are altered.
Slope Formula
The slope of a line is a measure of its steepness. It's a crucial concept that is determined by examining how much a line rises or falls as it moves horizontally across the graph. The basic slope formula is \(m = \frac{\Delta y}{\Delta x}\), which signifies the change in \(y\) over the change in \(x\).

For lines in the form \(y = mx + b\), \(m\) directly represents the slope. This slope tells us whether the line is slanting upwards, downwards, or is even perfectly horizontal.

Think of the slope as the **angle** the line makes with the horizontal:
  • Positive Slope: Line rises from left to right.
  • Negative Slope: Line falls from left to right.
  • Zero Slope: Line is flat, horizontal.
In this problem, increasing the x-intercept while keeping the y-intercept fixed alters the steepness without flipping the slope's sign.
Graphing Lines
Graphing lines on a coordinate plane visually represents their intercepts, slope, and overall behavior. Every line's graph is determined by its equation, and changes to this equation impact its graphical representation.

When you move a line on a graph while maintaining the same y-intercept but changing the x-intercept, you effectively adjust how steep the line is.
  • Graphical Shift: As you increase the x-intercept, the line stretches horizontally, which flattens it. The line becomes less steep compared to its original position.
  • Interpreting Changes: In terms of the slope, this adjustment means reducing the absolute value of a negative slope, making it less negative and closer to zero, though it still retains its negative nature unless further adjusted.
Visualizing this on a graph makes it easier to understand how minor movements along axes can shift slopes and intercepts, ultimately altering the line's character.

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