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

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

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 $y=0$).
• The y-intercept: where a line crosses the y-axis (i.e., where $x=0$).
• The slope: which measures the “steepness” or inclination of the line.

We will explore each of these ideas in detail.

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X-intercept

A line’s x-intercept is the point at which the line crosses the x-axis. At any point on the x-axis, $y=0.$

Finding the x-intercept from an equation

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

1. Substituting $y=0$ into the equation,
2. Solving for $x$.

Example (X-intercept from standard form) Suppose we have $4x+2y=8.$

1. Let $y=0$. The equation becomes $4x=8.$
2. Solve for $x$: $x=2.$

Therefore, the x-intercept is the point $(2,0)$.

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Y-intercept

A line’s y-intercept is the point at which the line crosses the y-axis. At any point on the y-axis, $x=0.$

Finding the y-intercept from an equation

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

1. Substituting $x=0$ into the equation,
2. Solving for $y$.

Example (Y-intercept from slope-intercept form) If the line is given by $y=3x+5$:

1. Let $x=0$. That gives $y=3(0)+5=5.$

Hence, the y-intercept is $(0,5)$.

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Slope of a line

The slope of a line measures how steep the line is and the direction it moves. For two points on a line, say $(x_1,y_1)$ and $(x_2,y_2)$, the slope $m$ is:

$m=\frac{y_2-y_1}{x_2-x_1}.$

If an equation is in the form $y=mx+b$, then $m$ (the coefficient of $x$) is the slope, and $b$ is the y-intercept.

Key slope types

• **Positive slope**: The line goes up as you move from left to right.
• **Negative slope**: The line goes down as you move from left to right.
• **Zero slope**: The line is perfectly horizontal ($m=0$).
• **Undefined slope**: The line is vertical ($x=\text{constant}$), and the slope formula would have a zero denominator.

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Examples

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

Let’s use the equation $y=-2x+6.$

1. **Y-intercept**: Let $x=0$. Then $y=6$. So y-intercept is $(0,6)$.
2. **X-intercept**: Let $y=0$. Then $0=-2x+6$ → $-2x=-6$ → $x=3$. So x-intercept is $(3,0)$.
3. **Slope**: The coefficient of $x$ is $-2$. So $m=-2$.

Example 2: Use two points to find slope

Suppose a line passes through $(2,5)$ and $(6,1)$. Find its slope.

• Using $m=\frac{y_2-y_1}{x_2-x_1}$:

$m=\frac{1-5}{6-2}=\frac{-4}{4}=-1.$

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

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Practice problems

1. For the equation $3x+y=9$:
   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 $(-2,7)$ and $(4,1)$. Find its slope. Is it positive, negative, or zero?
3. Given $y=5$, 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 $x=4$, what is its slope? Does it have a y-intercept?

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