Alg2H
5-3 Using the Discriminant, x-Intercepts, and the Quadratic Formula WK#6 (p. 1 of 6)
Lesson / Homework --Complete without calculator
Read p.181-p.186. Textbook required for reference as well as to check some answers
DETERMINING THE NUMBER OF X-INTERCEPTS
1. From Vertex Form: y – k = a(x – h)2
# of x-intercepts determined by the location of vertex and sign of “a”.
a) If k > 0 and a < 0, the vertex is ________ the x-axis and the graph goes ________.
If k < 0 and a > 0, the vertex is ________ the x-axis and the graph goes ________.
Both result in __________ x-intercept(s)
(How many?)
b) If k = 0 and a < 0, the vertex is ________ the x-axis and the graph goes ________.
If k = 0 and a > 0, the vertex is ________ the x-axis and the graph goes ________.
Both result in __________ x-intercept(s)
(How many?)
c) If k > 0 and a > 0, the vertex is ________ the x-axis and the graph goes ________.
If k < 0 and a < 0, the vertex is ________ the x-axis and the graph goes ________.
Both result in __________ x-intercept(s)
(How many?)
2. From General Form: y = ax2 + bx + c
# of x-intercepts determined by the ______________, D =_______________
x-intercepts are points where y = 0. Therefore they are the same as the roots of the equation:
0 = ax2 + bx + c
which can be solved by using the Quadratic formula: x = _______________________.
Which part of Quad formula determines the number and type of solutions? __________________
If D > 0, there are _______________ x-intercepts (or real roots).
(How many?)
If D is a perfect square, then they are _________________.
If D is not a perfect square, then they are ___________________.
If D = 0, there is _______________ x-intercept (or real root)
(How many?)
which is _____________ and the same as the ____________ of the function.
If D < 0, there are _______________ x-intercepts or real roots. (The roots are ________________.)
(How many?)
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SAMPLE PROBLEMS: (Complete without calculator)
Using General Form:
1) y = -5x2 + 5x +30
WK#6 (p. 2 of 6)
a) Quickly determine x-coordinate of vertex and use it to determine the y-coordinate of the vertex.
Remember: Quick way to determine Vertex: V(h,k) = V((________, f(_________))
How could you decide the # of x-intercepts from the information so far?
b) Use discriminant to determine the # of x-intercepts and if their value is rational or irrational:
c) Find the x-intercepts by factoring, if possible, or use quadratic formula if not possible.
State answers in exact simplified form. (Note: If irrational, radical form is the exact form.)
(How can the discriminant be used to decide if a quadratic is factorable? ___________________)
d) Sketch the graph of the function. Label coordinates of vertex, axis of symmetry, x-intercepts, y-intercept
and symmetric point to y-intercept.
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2. y = 2x2 – 8x + 2
(Complete without calculator)
WK#6 (p. 3 of 6)
a) Quickly determine x-coordinate of vertex and use it to determine the y-coordinate of the vertex.
How could you decide the # of x-intercepts from the information so far?
b) Use discriminant to determine the # of x-intercepts and if their value is rational or irrational:
c) Find the x-intercepts by factoring, if possible, or use quadratic formula if not possible.
State answers in exact simplified form. (Note: If irrational, radical form is the exact form.)
(How can the discriminant be used to decide if a quadratic is factorable? ___________________)
d) Sketch the graph of the function. Label coordinates of vertex, axis of symmetry, x-intercepts, y-intercept
and symmetric point to y-intercept.
3
3.
y = –16x2 + 40x – 25
(Complete without a calculator)
WK#6 (p. 4 of 6)
a) Quickly determine x-coordinate of vertex and use it to determine the y-coordinate of the vertex.
How could you decide the # of x-intercepts from the information so far?
b) Use discriminant to determine the # of x-intercepts and if their value is rational or irrational:
c) Find the x-intercepts by factoring, if possible, or use quadratic formula if not possible.
State answers in exact simplified form. (Note: If irrational, radical form is the exact form.)
(How can the discriminant be used to decide if a quadratic is factorable? ___________________)
d) Sketch the graph of the function. Label coordinates of vertex, axis of symmetry, x-intercepts, y-intercept
and symmetric point to y-intercept.
4
Using Vertex form,
Find the x-intercepts in exact simplified form and the coordinates of the vertex:
(Complete without calculator) (Refer to Wk#4, problem #6)
WK#6 (p. 5 of 6)
4. y + 32 = 25(x – 3)2
Using Intercept Form, sketch a graph of the function by
Finding the x-intercepts, the coordinates of the vertex, y-intercept and the symmetric pt:
(Refer to Wk#4, problem #5)
(Complete without calculator)
5. y = -7(x – 3)( x + 4)
Problems 6-7, follow these directions:
a) Transform each equation to vertex form by completing the square.
b) State the coordinates of the vertex and check by using the quick method.
6) y = 2x2 – 7x + 12
7) y = -3x2 – 4x + 5
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WK#6 (p. 6 of 6)
Problems 8-11, follow these directions:
a) From the general form, quickly determine the x-coordinate of the vertex and use it to determine the y-coordinate of the
vertex.
b) Find the value of the discriminant to determine the number of x-intercepts and if their value is rational or irrational.
c) If they exist, find the x-intercepts in “exact” simplified (radical) form.
Use factoring, if possible. Otherwise use the quadratic formula.
d) Sketch the graph of each function. Label coordinates of vertex, axis of symmetry, x and y-intercepts, if they exist and
symmetric point to y-intercept.
8) y= -4x2 – 8x + 12
(Answer similar to Exercise 5-3: #35 but y values multiplied by 4)
9) y = -4x2 + 4x – 1
(Check answer with Exercise 5-3: #39)
10) y = x2 + 2x + 5
11) y = x2 + 2x – 5
(Check answer with Exercise 5-3: #41)
(Check answer with Exercise 5-3: #43)
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