# 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.

## 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:
* 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 $4x=8.$
* 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 $x$ 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 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.

## Examples
### Example 1: Find 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 $3x+y=9$:
** Find the x-intercept.
** Find the y-intercept.
** 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?