Which graph represents the function $y-3=\frac{3}{2}(x-4)$?
Answer (2)
Rewrite the given equation y − 3 = 2 3 ( x − 4 ) in slope-intercept form: y = 2 3 x − 3 .
Identify the slope as 2 3 and the y-intercept as -3.
The graph is a line with a positive slope, crossing the y-axis at (0, -3).
The correct graph represents a line with slope 2 3 and y-intercept -3.
Explanation
Understanding the Equation The given equation is in point-slope form: y − 3 = 2 3 ( x − 4 ) . We need to identify the graph that represents this equation.
Converting to Slope-Intercept Form Let's rewrite the equation in slope-intercept form ( y = m x + b ) to easily identify the slope and y-intercept. Starting with y − 3 = 2 3 ( x − 4 ) , we distribute the 2 3 on the right side: y − 3 = 2 3 x − 2 3 ( 4 ) y − 3 = 2 3 x − 6 Now, add 3 to both sides to isolate y : y = 2 3 x − 6 + 3 y = 2 3 x − 3
Identifying Slope and Y-Intercept From the slope-intercept form, y = 2 3 x − 3 , we can identify the slope and y-intercept. The slope, m , is 2 3 , and the y-intercept, b , is -3. This means the line crosses the y-axis at the point (0, -3).
Analyzing the Graph The line has a positive slope of 2 3 , which means it goes upwards from left to right. For every 2 units we move to the right on the graph, we move 3 units up. The y-intercept is -3, so the line crosses the y-axis at (0, -3).
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you're tracking the cost of a taxi ride, the initial fee is the y-intercept, and the cost per mile is the slope. By knowing these values, you can predict the total cost of your ride. Similarly, in physics, the equation of motion for an object moving at a constant velocity is a linear equation, where the slope represents the velocity and the y-intercept represents the initial position.
To find the graph of the function y − 3 = 2 3 ( x − 4 ) , we convert it to slope-intercept form: y = 2 3 x − 3 . The slope is 2 3 , and the y-intercept is at (0, -3). The correct graph should represent a line rising from (0, -3) with a positive slope of 2 3 .
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