1. The roots of a quadratic equation are the values of x that make the related function zero.
The real roots are also the x-intercepts of the parabola. Look at the graph of y = x
2
– 4x + 3.
A. How many roots does x
...
1. The roots of a quadratic equation are the values of x that make the related function zero.
The real roots are also the x-intercepts of the parabola. Look at the graph of y = x
2
– 4x + 3.
A. How many roots does x
2
– 4x + 3 = 0 have? What are the roots?
B. Change c to 4.0. How many roots does x
2
– 4x + 4 = 0 have?
2. Now graph y = x
2
– 4x + 8 in the Gizmo, and look at the resulting parabola. Do you think
x
2
– 4x + 8 = 0 has any real roots? Explain.
3. Vary the values of a, b, and c. In general, how many real roots are possible for a quadratic
equation?
4. Graph y = x
2
+ 6x + 5. Turn on Show axis of symmetry x = –b/(2a). The axis of symmetry
is a line that divides a parabola into two halves that are mirror images.
A. How does the location of the axis of symmetry relate to the location of the two
x-intercepts?
B. Move the a, b, and c sliders. Which values affect the axis of symmetry?
C. The equation of the axis of symmetry is x =
a
b
2
. How does this explain what you
observed above?
5. Suppose you know the line of symmetry for a quadratic function.
A. From just this information, can you find the x-intercepts? Explain.
B. Suppose the axis of symmetry of the graph of a quadratic function is at x = 6. If one
root of the related quadratic equation is –1.5, what is the other root?
This study source was downloaded by 100000827637563 from CourseHero.com on 09-12-2021 15:37:11 GMT -05:00
https://www.coursehero.com/file/73552774/RootsQuadraticSEpdf/
This study resource was
shared via CourseHero.com
2019
Activity B:
The quadratic
formula
Get the Gizmo ready:
Be sure the CONTROLS and REAL PLANE tabs
are selected.
1. Some quadratic equations are difficult to factor. In these cases, you can use the quadratic
formula, x =
a
b b ac
2
4
2
, to find the roots of the quadratic equation ax2
+ bx + c = 0.
A. Graph y = 3x
2
– x – 4. Select the SOLUTION tab to see how the quadratic formula is
used to find the roots of 3x
2
– x – 4 = 0. What are the roots?
Click on the CONTROLS tab to check that these are the x-intercepts of the graph.
B. Use the quadratic formula to find the
roots of 2x
2
+ x – 10 = 0. Show your
work in the space to the right. Then
check your answer in the Gizmo.
2. The discriminant is the part of the quadratic formula that is under the radical, b
2
– 4ac. It
provides useful information about the number of real roots of a quadratic equation. On the
CONTROLS tab, turn on Show discriminant computation.
A. Graph each quadratic
function listed to the
right in the Gizmo.
Then state the number
of real roots of the
related equation
(ax2
+ bx + c = 0) and
give the discriminant.
B. Vary a, b, and c, and watch how the number of real roots and the discriminant
change. In general, how does the discriminant relate to the number of real roots?
C. Why do you think the discriminant determines the number of real roots of a quadratic
equation?
(Activity B continued on next page)
Function Number of real roots Discriminant
y = x
2
+ 6x + 9
y = x
2
– 5x – 8
y = x
2
– 4x + 6
This study source was downloaded by 100000827637563 from CourseHero.com on 09-12-2021 15:37:11 GMT -05:00
https://www.coursehero.com/file/73552774/RootsQuadraticSEpdf/
This study resource was
shared via CourseHero.com
2019
Activity B (continued from previous page)
3. A fraction with a sum in the numerator can be rewritten as the sum of two fractions with the
same denominator. For example,
2
1 x
=
2
1
+
2
x
.
A. In the space to the right, rewrite the quadratic
formula as the sum of two fractions.
B. What line does the first fraction describe?
C. What is the second numerator the same as?
D. Whenever a quadratic function has two x-intercepts, they are always equidistant from
the axis of symmetry. Use your findings from above to explain why this is true.
[Show More]
