which graph represents the function $y - 3 = \\frac{3}{2}(x - 4)$?
which graph represents the function $y - 3 = \\frac{3}{2}(x - 4)$?
Answer
Explanation:
Step1: Rewrite the equation in slope - intercept form
The given equation is (y - 3=\frac{3}{2}(x - 4)). We use the distributive property (a(b - c)=ab - ac) (here (a = \frac{3}{2}), (b=x), (c = 4)) to expand the right - hand side: (y-3=\frac{3}{2}x-6) Then, we add 3 to both sides of the equation to solve for (y): (y=\frac{3}{2}x-6 + 3) (y=\frac{3}{2}x-3)
Step2: Identify the slope and y - intercept
In the slope - intercept form of a linear equation (y=mx + b), where (m) is the slope and (b) is the y - intercept. For the equation (y=\frac{3}{2}x-3), the slope (m=\frac{3}{2}) (which is positive, so the line should be increasing from left to right) and the y - intercept (b=- 3) (the line crosses the y - axis at the point ((0,-3))).
Step3: Analyze the graphs
- The first graph: Check the slope and y - intercept. The line has a positive slope and crosses the y - axis at a negative value. Let's verify the slope. If we take two points on the line, say from the graph, when (x = 2), (y=0) (since it crosses the x - axis at (x = 2)) and when (x=0), (y=-3). The slope between ((0,-3)) and ((2,0)) is (m=\frac{0 - (-3)}{2-0}=\frac{3}{2}), which matches our calculated slope.
- The second graph: It has a negative slope (decreasing from left to right), so it can't be the graph of (y=\frac{3}{2}x-3) (since our slope is positive).
- The third graph: It crosses the y - axis at (y = 0), but our y - intercept is (-3), so it is not the correct graph.
- The fourth graph: It has a negative slope, so it can't be the graph of our function (since our slope is positive).