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Find Slope, Y-intercept and Linear Equation from a Graph
Mathematics
Grade 8 of Junior High School
Question Content
Given the graph of a linear line, find: 1) the slope (m), 2) the y-intercept (b), 3) the equation of the line in slope-intercept form (y = mx + b)
Correct Answer
m = -1/2, b = -2, y = -1/2x - 2
Detailed Solution Steps
1
Step 1: Identify two clear points on the line. From the graph, we can use (-4, 0) and (0, -2), or (-4, 0) and (4, -4).
2
Step 2: Calculate the slope (m) using the formula m = (y₂ - y₁)/(x₂ - x₁). Using points (-4, 0) and (0, -2): m = (-2 - 0)/(0 - (-4)) = -2/4 = -1/2.
3
Step 3: Find the y-intercept (b). The y-intercept is the y-value where the line crosses the y-axis. From the graph, the line crosses (0, -2), so b = -2.
4
Step 4: Write the equation in slope-intercept form y = mx + b. Substitute m = -1/2 and b = -2, so the equation is y = -1/2x - 2.
Knowledge Points Involved
1
Slope of a Linear Line
The slope (m) measures the steepness of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points (x₁,y₁) and (x₂,y₂) using the formula m = (y₂ - y₁)/(x₂ - x₁). A negative slope means the line decreases from left to right.
2
Y-intercept of a Linear Line
The y-intercept (b) is the point where the line crosses the y-axis, which occurs when x=0. It represents the initial value of the dependent variable y when the independent variable x is 0.
3
Slope-Intercept Form of a Linear Equation
The slope-intercept form is written as y = mx + b, where m is the slope and b is the y-intercept. This form is used to quickly graph a line and understand its key characteristics: steepness and starting point on the y-axis.
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