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!