## 1. Topic Overview

**Main Concept/Theme**: Understanding and finding intercepts of a line in the coordinate plane.

**Key Learning Objectives**:

– Understand what intercepts are in the context of graphs.

– Learn how to find x-intercepts and y-intercepts of a given linear equation.

– Apply the knowledge to solve problems involving intercepts.

## 2. Key Terms and Definitions

**Intercept**: The point where a graph intersects an axis.**x-intercept**: The point where the graph intersects the x-axis (where ( y = 0 )).**y-intercept**: The point where the graph intersects the y-axis (where ( x = 0 )).**Linear equation**: An equation that makes a straight line when it is graphed. Typically in the form ( y = mx + b ) or ( ax + by = c ).

## 3. Main Content Sections

### 3.1 Understanding Intercepts

#### x-Intercept

- The x-intercept is the point where the graph crosses the x-axis.
- At the x-intercept, the value of ( y ) is always 0.
- To find the x-intercept, set ( y ) to 0 in the equation and solve for ( x ).

#### y-Intercept

- The y-intercept is the point where the graph crosses the y-axis.
- At the y-intercept, the value of ( x ) is always 0.
- To find the y-intercept, set ( x ) to 0 in the equation and solve for ( y ).

### 3.2 Finding Intercepts from Equations

#### Example 1: ( y = 2x + 3 )

**y-Intercept**: Set ( x = 0 )

[

y = 2(0) + 3

]

[

y = 3

]

The y-intercept is ( (0, 3) ).**x-Intercept**: Set ( y = 0 )

[

0 = 2x + 3

]

[

2x = -3

]

[

x = -\frac{3}{2}

]

The x-intercept is ( \left( -\frac{3}{2}, 0 \right) ).

#### Example 2: ( 3x – 4y = 12 )

**y-Intercept**: Set ( x = 0 )

[

3(0) – 4y = 12

]

[

-4y = 12

]

[

y = -3

]

The y-intercept is ( (0, -3) ).**x-Intercept**: Set ( y = 0 )

[

3x – 4(0) = 12

]

[

3x = 12

]

[

x = 4

]

The x-intercept is ( (4, 0) ).

## 4. Example Problems or Case Studies

### Problem 1

Find the intercepts of the equation ( y = -3x + 6 ).

**Solution**:

– **y-Intercept**: Set ( x = 0 )

[

y = -3(0) + 6 = 6

]

The y-intercept is ( (0, 6) ).

**x-Intercept**: Set ( y = 0 )

[

0 = -3x + 6

]

[

3x = 6

]

[

x = 2

]

The x-intercept is ( (2, 0) ).

## 5. Summary or Review Section

- Intercepts are where a graph meets the axes.
- The x-intercept occurs where ( y = 0 ); the y-intercept occurs where ( x = 0 ).
- To find intercepts, substitute the relevant value (0) into the equation and solve for the other variable.

## 6. Self-Assessment Questions

- Find the x and y intercepts of the equation ( y = \frac{1}{2}x + 4 ).
- What is the y-intercept of the equation ( 5x – 2y = 10 )?
- If the x-intercept of a line is ( 7 ) and the y-intercept is ( -5 ), write down these points in coordinate form.

## 7. Connections to Other Topics/Subjects

**Coordinate Geometry**: Intercepts are fundamental in understanding the shapes and positions of graphs.**Algebra**: Solving equations is a key skill to finding intercepts.**Real-World Applications**: Understanding intercepts helps in predicting outcomes, such as where a trend line might cross a target value in data analysis.

**Remember** to practice regularly and seek help if a concept isn’t clear. Keep testing your understanding and soon, finding intercepts will become second nature!

Good luck with your studies!