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Find Y-Intercept, Slope and Linear Equation from a Graph
Mathematics
Grade 8 (Junior High School)
Question Content
Use the graph to answer the following questions: 1. What is the y-intercept (b)? 2. What is the slope (m)? 3. Write an equation for the line on the graph (y = mx + b)
Correct Answer
1. b = 3; 2. m = 1/2; 3. y = (1/2)x + 3
Detailed Solution Steps
1
Step 1: Identify the y-intercept (b). The y-intercept is the point where the line crosses the y-axis. Looking at the graph, the line crosses the y-axis at (0, 3), so b = 3.
2
Step 2: Calculate the slope (m). Use the slope formula m = (y₂ - y₁)/(x₂ - x₁). Select two clear points on the line: (0, 3) and (2, 4). Plug the values into the formula: m = (4 - 3)/(2 - 0) = 1/2. Alternatively, use the rise-over-run method: from (0,3), moving up 1 unit and right 2 units reaches the next point on the line, so rise/run = 1/2 = m.
3
Step 3: Write the linear equation. Substitute m = 1/2 and b = 3 into the slope-intercept form y = mx + b, resulting in y = (1/2)x + 3.
Knowledge Points Involved
1
Y-intercept of a Linear Line
The y-intercept (denoted as b) is the coordinate point where a straight line crosses the y-axis of a coordinate plane. In the slope-intercept form y = mx + b, it is the constant term, representing the value of y when x = 0. It is used to define the starting position of the line on the vertical axis.
2
Slope of a Linear Line
The slope (denoted as m) measures the steepness and direction of a straight line. It is calculated as the ratio of the vertical change (rise, Δy) to the horizontal change (run, Δx) between any two points (x₁,y₁) and (x₂,y₂) on the line, using the formula m = (y₂ - y₁)/(x₂ - x₁). A positive slope means the line rises from left to right, while a negative slope means it falls.
3
Slope-Intercept Form of a Linear Equation
The slope-intercept form is written as y = mx + b, where m is the slope of the line and b is the y-intercept. This form is widely used to quickly graph a line, identify its key features, and solve problems involving linear relationships in real-world scenarios, such as rate of change problems.
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