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

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.

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 terms of $x$ and $y$ , you can find its x-intercept by:

Substituting $y=0$ into the equation
Solving for $x$

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

Let $y=0$ . The equation becomes $4x+2(0)=8.$ This is the same as $4x=8.$
Solve for $x$ : $x=2.$

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

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 terms of $x$ and $y$ , you can find its y-intercept by:

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

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

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

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

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 slope-intercept 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 (i.e. the $y$ value increases as the $x$ value increases)
**Negative slope**: The line goes down as you move from left to right (i.e. the $y$ value decreases as the $x$ value increases).
**Zero slope**: The line is perfectly horizontal ( $m=0$ ). This also means that all $y$ values are 0.
**Undefined slope**: The line is vertical ( $x={\text{constant}}$ ), and the slope formula would have a zero denominator.

Examples

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

Let’s use the equation $y=-2x+6.$

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

Practice Problems

For the equation $3x+y=9$ $3x+y=9$ :
1. Find the x-intercept.
2. Find the y-intercept.
3. Rewrite the equation in slope-intercept form to identify the slope.
A line goes through the points $(-2,7)$ and $(4,1)$ . Find its slope. Is it positive, negative, or zero?
Given $y=5$ , is there an x-intercept? Explain what the graph looks like and how that relates to slope.
Bonus: If a line is described by $x=4$ , what is its slope? Does it have a y-intercept?

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Algebra 1

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