Download The Discriminant

Worksheet: Discriminant of Quadratic Function
Nicholas Bennett & LFS
The Discriminant
Recall that the definition of a quadratic function in standard form is: f (x)  ax 2  bx  c , a  0 .
A quadratic function can have zero, one, or two real roots. Only real roots can be graphed!
Roots can also be called zeros or x-intercepts. They are x-values that make f(x)=0.
That is, roots are solutions to the equation ax 2  bx  c  0 .
The discriminant is a nifty calculation that tells us the number of roots of a quadratic function.
If they exist, we can find the roots of this function with the formula:
b  b2  4ac
b  b2  4ac
. This means: x1 
x1 , x2 
2a
2a
b  b2  4ac
.
x2 
2a
The reason we write a single formula with a plus/minus sign is so that we do the simplifications for
the terms b , b2  4ac and 2a only once! When we are done with simplifying the plus/minus
formula, then we separate into two formulas: (1) for x1 with a + and (2) for x2 with a –.
The term underneath the radical sign is special. It is called the discriminant.
The discriminant is the expression: b2  4ac . It is the same for both x1 and x1.
Mini review.
The graph of a quadratic function is a parabola.
The characteristic points of a quadratic function are the y-intercept, the roots (if they exist) and
the vertex (the maximum or minimum point). With these points we can sketch the graph.
The coefficient a of x2 determines the direction of the parabola.
If a>0, the parabola opens up. If a<0, the parabola opens down.
The y-intercept is y=c. (The y-intercept lies on the y-axis, that is, the line x=0.)
Parabolas are symmetric. Do not confuse the y-intercept with the vertex and skew your parabola.
Remember – we can only graph points whose coordinates are real numbers. Here we do not care
at all about imaginary numbers. For us, they are undefined and do not exist!
Do not confuse “ugly” numbers: 2  5  4.2 with numbers that do not exist: 1  4 . We can
graph ugly numbers by just rounding them to 1 decimal. We cannot graph 1  4 .
Algebra 2
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Worksheet: Discriminant of Quadratic Function
Nicholas Bennett & LFS
Investigation
Open discriminant.ggb.
Notice the graph of f(x) and that the coefficients a, b, and c of f(x) are sliders Experiment by
changing the sliders for a, b, and c. You should already be familiar with the affects of changing the
coefficients of f(x).
Q1.
When you finish, set the slider values to match the coefficients in the quadratic function below:
f (x)  x 2  4 x  5 . How many times does the function y = f(x) cross the x-axis?
________________________________________________________________________
Q2.
Zoom out (mouse scroll button). Using your eyes, find the x-value of these points, the y-value of the
y-intercept, the coordinates of the vertex and then sketch the function below at left.
Q3.
On the applet, find the value of the discriminant. Write this down. Verify this by calculating the
value of the discriminant by hand.
_________________________________________________________________________
Q4.
Select the “Show roots” box. Write down the values for x1 and x2. Check to see if these roots are
zeros by plugging each value back into y  x 2  4 x  5 .
x1 =
. Check:
____________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________
x2 =
. Check:
_____________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________
== Set the sliders so that the function being displayed is f (x)  2x 2  7x  5 .
Q5.
Looking at the graph, how many solutions are there to the equation 2 x 2  7 x  5  0 ? Calculate
the value of the discriminant. Does it match the value of the discriminant shown in the applet?
Find the characteristic points (y-intercept, roots and vertex) and sketch the graph above right.
Algebra 2
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Worksheet: Discriminant of Quadratic Function
Q6.
Nicholas Bennett & LFS
We have seen two cases when the discriminant is greater than 0 and there are two zeros. Will this
always be the case? Experiment by changing sliders a, b, and c such that the discriminant in always
positive. Record your results below. Find an example where a<0.
Discriminant =
>0 .
Quadratic function:
. Number of zeros:
.
_____________________________________________________________________________________________________________________________________________________________________________________________________________ __________________________
Discriminant =
>0 .
Quadratic function:
. Number of zeros:
.
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Discriminant =
>0 .
Quadratic function:
. Number of zeros:
.
__________________________________________________________________________________________________________________________________________________________________________________________________________ _____________________________
Q7.
In summary, when will a quadratic function have two zeros (roots)?
____________________________________________________________________________
Q8.
Adjust sliders a, b, and c so that the discriminant is negative. Sketch the function below left.
Q9.
How many roots does this quadratic function have? Write down all of this information below.
Discriminant =
>0 .
Quadratic function:
. Number of zeros:
.
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Q10. Explain how you can tell graphically that a quadratic function has no roots.
___________________________________________________________________________
Q11. Adjust sliders a, b, and c to show another negative discriminant. Sketch the function below right.
Do you get the same answers to Q11 and Q12?
Discriminant =
>0 .
Quadratic function:
. Number of zeros:
.
_____________________________________________________________________________________________________________________________________________________________________________________________________________ __________________________
Algebra 2
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Worksheet: Discriminant of Quadratic Function
Nicholas Bennett & LFS
Q12. Make a conjecture about the number of solutions of a quadratic when the discriminant is 0.
_____________________________________________________________________________
==Set the sliders to match the function shown: f (x)  x 2  2x  1
Q13. What is the value of the discriminant? How many roots does this function have?
Discriminant =
>0 .
Number of zeros:
.
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Q14. Is this true for all quadratic functions when the discriminant is 0?
== Adjust sliders a, b, and c to show the following function: f (x)  x 2  6x  9
Q15. Does the quadratic function cross the x-axis? Based on previous questions, how many solutions
does this quadratic function have?
__________________________________________________________________________________________
Closure
The roots of a quadratic function f(x) are the x-intercepts. They are the solution to f (x)  0 . We
can conclude that the y-coordinate for all roots is 0. That is, the roots of an equation such as
y  2x 2  4 x  5 are the solutions to 2 x 2  4 x  5  0 .
Q16. Without graphing, find the value of discriminant, its sign and use this to determine the number of
real solutions to the following equations.
Equation
Algebra 2
Coefficients
Discriminant
4
Sign
Number of
Worksheet: Discriminant of Quadratic Function
Nicholas Bennett & LFS
solutions
x2 + 8x + 6=0
a=
b=
c=
0 = -5x2 + 4x - 1
a=
b=
c=
2x2+2x+0.5=0
a=
b=
c=
0 = 2x-x2 + 2
a=
b=
c=
x 2  6 x  13
a=
b=
c=
x2=3
a=
b=
c=
2x-x2=0
a=
b=
c=
Q17. Describe how the value of the discriminant affects the type of rootss a quadratic function has:
Given a quadratic function f(x)=ax2+bx+c
Discriminant
Number of Roots
2
b2  4ac  0
0
Challenger questions:
Q18. What can you say about the vertex when the discriminant is 0?
_______________________________________________________________________________
Q19. You are a teacher and you quickly want to write down a quadratic function that opens up and has
two zeros. You don’t care if the zeros are whole numbers, but you want two zeros.
I say: It doesn’t matter what b is, it only matters that a>0 and c<0.
Using the applet, test this conjecture. Why do you think this is true? Explain your answer using the
discriminant and explain your answer using the graph (remember c is the y-intercept).
Explanation 1: If a>0 and c<0, then the discriminant is: : <0
=0
>0
(cross out incorrect)
Conclusion: The function has 2 zeros.
Explanation 2: If a>0 then the parabola: opens up opens down and if c<0 the y-intercept is above
below the x-axis. This means that the graph of this function crosses does not cross the
x-axis at 0 1 2 points.
Conclusion: The function has 2 zeros.
Make a several different sketches of a “normal” sized parabolas that open up (a=1) with c=-2 .
Algebra 2
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Worksheet: Discriminant of Quadratic Function
Nicholas Bennett & LFS
Check that they all have 2 roots.
Q20. Is the “opposite” true? That is, if a>0 but now c>0, is it always true that the quadratic function will
have no zeros? Using the applet, find three quadratic functions with different types of zeros and
write down the functions below, their determinant and the number of roots.
Quadratic Function with a>0 and c>0 Discriminant
Number of Roots
f(x)=
2
f(x)=
1
f(x)=
0
REAL CHALLENGERS a<0
Q21. If a<0 and c>0, then the quadratic function has two zeros. Why?
Q22. If a<0 and c>0, then the discriminant is: <0
=0
>0
(cross out incorrect).
If a<0 then the parabola: opens up opens down and if c>0 the y-intercept is above
the x-axis. This means that the graph of this function crosses
0
1
below
does not cross the x-axis at
2 points.
Q23. Finally, assume that a<0 and c<0. Using the applet, find three quadratic functions with different
types of zeros and write down the functions below, their determinant and the number of roots.
Quadratic Function with a<0 and c<0 Discriminant
Algebra 2
Number of Roots
f(x)=
2
f(x)=
1
f(x)=
0
6
Worksheet: Discriminant of Quadratic Function
Algebra 2
Nicholas Bennett & LFS
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