b How does the rate of change between a line and a parabola compare?
A
a 2 is the slope of the line.
Diagram:
B
b No, see solution.
Practice makes perfect
a Examining the functions we see that we have one linear equation and two quadratic equations.
Linear:& y=2x+5 [0.7em]
Quadratic:& y=2x^2+5 [0.3em]
Quadratic:& y=1/2x^2+5
To graph the linear equation, we need at least two points. The first point, we can find by plotting its y-intercept, (0,5). To plot a second point, we can use the function's slope which is what the 2 represents.
Let's continue by drawing the graph of the first quadratic. For this purpose, we need to make a table of values.
|c|c|c|
[-0.8em]
x & 2x^2+5 & y [0.2em]
[-0.8em]
-2 & 2( -2)^2+5 & 13 [0.2em]
[-0.8em]
-1 & 2( -1)^2+5 & 7 [0.2em]
[-0.8em]
0 & 2( 0)^2+5 & 5 [0.2em]
[-0.8em]
1 & 2( 1)^2+5 & 7 [0.2em]
[-0.8em]
2 & 2( 2)^2+5 & 13 [0.2em]
Let's mark these points in our diagram and draw a parabola through them.
Let's continue by drawing the last quadratic. For this purpose, we need to make another table of values.
|c|c|c|
[-0.8em]
x & 1/2x^2+5 & y [0.8em]
[-0.8em]
-4 & 1/2( -4)^2+5 & 13 [0.8em]
[-0.8em]
-2 & 1/2( -2)^2+5 & 7 [0.8em]
[-0.8em]
0 & 1/2( 0)^2+5 & 5 [0.8em]
[-0.8em]
2 & 1/2( 2)^2+5 & 7 [0.8em]
[-0.8em]
4 & 1/2( 4)^2+5 & 13 [0.8em]
Finally, we will plot the function's graph by marking the points in the diagram and connecting them with a smooth curve.
Examining the diagram, we notice that all three graphs intersect the y-axis at y=5. We also see that the red function intersect the x-axis at x=-2.5.
b The rate of change of a line is also known as the line's slope. The difference between a line's and a parabola's rate of change is that for a line it's constant. For a parabola, the rate of change depends on which two points you are using to measure it.
