Notice that for each argument x, the value of the latter function is opposite that of the previous one.
A
a
Graph:
x-intercepts: (3,0) and (- 52,0)
B
b
See solution.
Practice makes perfect
a
We will begin by graphing the given parabola. While graphing, we will find the x -intercepts and write these points in (x,y) form. To make a graph of the parabola, we will follow five steps.
Recall the intercept form of a quadratic function.
y=a(x-p)(x-q)
In this form where a ≠0, the x-intercepts are p and q. Let's consider the intercept form of our function.
y=2(x+5/2)(x-3)
⇕
y= 2(x-( -5/2))(x- 3)
We can see that a= 2, p= - 52, and q= 3. Therefore, the x-intercepts occur at ( - 52,0) and ( 3,0).
Find and Graph the Axis of Symmetry
The axis of symmetry is halfway between (p,0) and (q,0). Since we know that p= - 52 and q= 3, the axis of symmetry of our parabola is halfway between (- 52,0) and (3,0).
x=p+ q/2 ⇒ x=- 52+ 3/2=1/4
We found that the axis of symmetry is the vertical line x= 14.
Find and Plot the Vertex
Since the vertex lies on the axis of symmetry, its x-coordinate is 14. To find the y-coordinate, we will substitute 14 for x in the given equation.
The y-coordinate of the vertex is - 15 18. Therefore, the vertex is the point ( 14,- 15 18).
Draw the Parabola
Finally, we will draw the parabola through the vertex and the x-intercepts.
b
We want to find what changes and what stays the same once we change the function from Part A into the new one. We also want to state whether we can tell each of these things without graphing the function. First, notice that for each x the value of the latter function is opposite.
y= -( 2x^2-x-15)
⇕
y = -2x^2 +x + 15
This means that to find the graph of the new function we just need to reflect the previous graph across the x-axis. Also, note that the obtained function still is quadratic, so its graph is still a parabola.
As we can see, while the original function was a parabola that opens upwards, now the parabola opens downwards. Previously, the y-coordinate of the vertex told us the minimal value of a function, but now the new coordinate tells us the maximum value of the new function.
Different Properties
Property
Old function
New function
Direction of opening
Opens upwards
Opens downwards
Vertex
( 14, -15 18)
( 14, 15 18)
Has a minimum
✓
*
Has a maximum
*
✓
We can find these properties without graphing. Recall that parabola opens upwards if the leading coefficient of a function is positive, and downward if it is negative. Previously, it equaled 2, a positive number, so the parabola opened upwards. Now, however, it equals -2, which is negative. The parabola now opens downward.
y = -( 2x^2-x-15)
⇕
y = -2x^2+x+15
The quadratic function has a minimum if its graph opens upwards, and maximum if it opens downwards. Let's explain the second property. The new function is a reflection across the x-axis so the vertex of the old function will become the vertex of the new one, and its y-coordinate will change signs.
( 14, -15 18)
⇔
( 14, 15 18)
Note that some properties of our graph stayed the same. Among these, we have the same x-intercepts and the same axis of symmetry. Also, we already know that the graphs of both functions are parabolas.
Now, since the graph of the new function is the reflection across the x-axis of the original function, we can tell each of these information without graphing. Since x-intercepts lie on the x-axis, they will not change during such reflection. Similarly, the axis of symmetry passes through their midpoint, so it will also stay the the same.
