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Alg2H 5-3 Using the Discriminant, x-Intercepts, and the Quadratic Formula WK#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 (ROOTS)

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 _______________ real x-intercepts (or 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 _______________ real x-intercept (or root) (How many?) which is _____________ and the same as the ____________ of the function.

If D < 0, there are _______________ real x-intercepts or roots. (The roots are ________________.) (How many?)

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SAMPLE PROBLEMS: (Complete without calculator) WK#6

Using General Form:

1) y = -5x2 + 5x +30

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

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.

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3. y = –16x2 + 40x – 25 (Complete without a calculator) WK#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.

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Using Vertex form, WK#6

Find the x-intercepts in exact simplified form and the coordinates of the vertex:

(Complete without calculator) (Refer to Wk#4, problem #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:

(Complete without calculator) (Refer to Wk#4, problem #5)

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 = -3x

2 – 4x + 5

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Problems 8-11, follow these directions: WK#6 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 9) y = -4x

2 + 4x – 1

(Answer similar to Exercise 5-3: #35 but y values multiplied by 4) (Check answer with Exercise 5-3: #39)

10) y = x2 + 2x + 5 11) y = x

2 + 2x – 5

(Check answer with Exercise 5-3: #41) (Check answer with Exercise 5-3: #43)