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Algebra 1/Unit 6: X-intercepts and Y-intercepts, Slopes

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Algebra 1/Unit 6: X-intercepts, Y-intercepts, and Slopes

Welcome to Week 6 of the Algebra 1 Course. This week, we will focus on:

  • What it means to find the x-intercept(s) and y-intercept(s) of a linear equation
  • How to identify the slope of a line (using a graph or algebraic formula)
  • Tips on interpreting these concepts for straight-line equations

Introduction

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By now, you know how to plot linear equations by finding a few points and drawing a line. However, there are more straightforward techniques for specific tasks, like finding:

  • The x-intercept: where a line crosses the x-axis (i.e., where ).
  • The y-intercept: where a line crosses the y-axis (i.e., where ).
  • The slope: which measures the “steepness” or inclination of the line.

We will explore each of these ideas in detail.


X-intercept

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A line’s x-intercept is the point at which the line crosses the x-axis. At any point on the x-axis,

Finding the x-intercept from an equation

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If you have a linear equation in terms of and , you can find its x-intercept by:

  1. Substituting into the equation
  2. Solving for

Example (X-intercept from standard form) Suppose we have

  1. Let . The equation becomes This is the same as
  2. Solve for :

Therefore, the x-intercept is the point .


Y-intercept

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A line’s y-intercept is the point at which the line crosses the y-axis. At any point on the y-axis,

Finding the y-intercept from an equation

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If you have a linear equation in terms of and , you can find its y-intercept by:

  1. Substituting into the equation
  2. Solving for .

Example (Y-intercept from slope-intercept form) If the line is given by :

  1. Let . That gives

Hence, the y-intercept is .


Slope of a Line

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The slope of a line measures how steep the line is and the direction it moves. For two points on a line, say and , the slope is:

If an equation is in the slope-intercept form , then (the coefficient of ) is the slope, and is the y-intercept.

Key Slope Types

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  • **Positive slope**: The line goes up as you move from left to right (i.e. the value increases as the value increases)
  • **Negative slope**: The line goes down as you move from left to right (i.e. the value decreases as the value increases).
  • **Zero slope**: The line is perfectly horizontal (). This also means that all values are 0.
  • **Undefined slope**: The line is vertical (), and the slope formula would have a zero denominator.

Examples

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Example 1: Find the x-intercept, y-intercept, and slope

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Let’s use the equation

  1. **Y-intercept**: Let . Then . So y-intercept is .
  2. **X-intercept**: Let . Then . So x-intercept is .
  3. **Slope**: The coefficient of is . So .

Example 2: Use two points to find slope

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Suppose a line passes through and . Find its slope.

  • Using :

Therefore, the slope is , indicating the line goes down 1 unit for every 1 unit you move to the right.


Practice Problems

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  1. For the equation :
    1. Find the x-intercept.
    2. Find the y-intercept.
    3. Rewrite the equation in slope-intercept form to identify the slope.
  2. A line goes through the points and . Find its slope. Is it positive, negative, or zero?
  3. Given , is there an x-intercept? Explain what the graph looks like and how that relates to slope.
  4. Bonus: If a line is described by , what is its slope? Does it have a y-intercept?