Graph Point Slope Form - Algebra-1

1. Fundamental Concepts

Definition: The point-slope form of a linear equation is given by $$y - y_1 = m(x - x_1)$$ , where $$m$$ is the slope and $$(x_1, y_1)$$ is a point on the line.
Slope: The slope $$m$$ represents the rate of change of $$y$$ with respect to $$x$$ .
Graphing: To graph a line using the point-slope form, plot the point $$(x_1, y_1)$$ and use the slope to find another point.

2. Key Concepts

Basic Rule: $$y - y_1 = m(x - x_1)$$

Slope Interpretation: The slope $$m$$ indicates the steepness and direction of the line.

Application: Used in various fields such as physics, economics, and engineering for modeling linear relationships.

3. Examples

Example 1 (Basic)

Problem: Graph the line that passes through the point $$(2, 3)$$ with a slope of $$4$$ .

Step-by-Step Solution:

Write the point-slope form: $$y - 3 = 4(x - 2)$$
Simplify if necessary: $$y - 3 = 4x - 8$$
Plot the point $$(2, 3)$$ and use the slope to find another point. From $$(2, 3)$$ , move up $$4$$ units and right $$1$$ unit to get $$(3, 7)$$ .

Validation: Substitute $$x = 3$$ into the equation: $$y - 3 = 4(3 - 2) \Rightarrow y - 3 = 4 \Rightarrow y = 7$$ . The point $$(3, 7)$$ lies on the line. ✓

Example 2 (Intermediate)

Problem: Graph the line that passes through the points $$(1, 5)$$ and $$(4, 9)$$ .

Step-by-Step Solution:

Calculate the slope: $$m = \frac{9 - 5}{4 - 1} = \frac{4}{3}$$
Use the point-slope form with either point. Using $$(1, 5)$$ : $$y - 5 = \frac{4}{3}(x - 1)$$
Plot the point $$(1, 5)$$ and use the slope to find another point. From $$(1, 5)$$ , move up $$4$$ units and right $$3$$ units to get $$(4, 9)$$ .

Validation: Substitute $$x = 4$$ into the equation: $$y - 5 = \frac{4}{3}(4 - 1) \Rightarrow y - 5 = 4 \Rightarrow y = 9$$ . The point $$(4, 9)$$ lies on the line. ✓

4. Problem-Solving Techniques

Visual Strategy: Use graph paper to accurately plot points and draw lines.
Error-Proofing: Double-check calculations and slopes by substituting points back into the equation.
Concept Reinforcement: Practice converting between different forms of linear equations to reinforce understanding.