Answer:
Graph 4
Step-by-step explanation:
When being asked problems like this one, even before working out any specifics, looking at the direction the line is moving in, you can at least eliminate some answers. When a graph has a negative slope, it will be decreasing as it moves to the right on the x-axis (its direction will look somewhat like a \, but not always this steep). Only two answers to this question have these properties, Graph 3 and Graph 4. Because of this trick, we're able to eliminate Graph 1 and Graph 2 as possible answers to this question, but how do we decide between Graph 3 and Graph 4? This is where knowledge on Linear Equations come into play.
- [tex]y=mx+b[/tex] is the parent function for linear equations, and its properties are as follows:
- [tex]m[/tex] is the equation's slope; determining the magnitude and direction of a linear line. Slope represents a ratio of how many units a function moves up or down to how much it moves left or right. For example, the equation [tex]y=\frac{2}{3}x[/tex] has a slope of [tex]\frac{2}{3}[/tex]. If the numerator of the function, [tex]2[/tex], represents how far the function moves upwards, and the denominator of the function, [tex]3[/tex], represents its movement to the right, then for every 2 units the function goes upwards, that's 3 units to the right. If there were negative values in the slope (ex: [tex]-\frac{2}{3}[/tex]), then one of the directions would be down, or left.
- [tex]b[/tex] represents the linear function's y-intercept; the y value it hits on the y-axis. For example, the equation [tex]y=x+1[/tex] will hit the y axis at the point (0,1). This will help us finish solving the equation.
Looking at the two remaining graphs and using our knowledge on [tex]b[/tex], we can see that Graph 3 hits the y-axis above [tex]0[/tex] and Graph 4 hits the y-axis below [tex]0[/tex]. If this problem is asking for the graph to have the same y-intercept as [tex]y=\frac{2}{3}x-2[/tex], then with [tex]b[/tex] being [tex]-2[/tex], we can figure that the function we want to make will have to hit the y-axis at [tex]-2[/tex], below [tex]0[/tex]. Only Graph 4 has these properties, so it is the answer.
