Using The Quadratic Formula to Sketch Graphs PDF

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Math 154B Solving Using the Quadratic Formula Worksheet The Quadratic Formula:

For quadratic equations: ax 2 bx c

x

b

b 2 4ac 2a

Solve each equation using the Quadratic Formula. 1. 4 x2 1x 20 0

3.

x2

3x

5.

x2

x

7. 4 x 2 7 x

0,

x2

x 24

4. x 2

x

2.

6. 4 x 2

5

0

8.

x2

0

x

x

0 0

9.

x2

x

11. x 2 2 x 48

13. 5 x2

x

0

10. 2 x2

23

2 12. 2 x

9

14. 5 x2

0x

4x

8x

25

Answers:

5 ,x 4 x 8, x

1. x

7. x

2.

8. x

3. x 4. x 5. x 6. x

3

21 2 5 5 2 1 5 2 2 5 2

9. x 10. x

5 ,x 4 2, x 1

13 2

7

3

2 11. x 8, x

12. x

9

3

2 13. x = not a real number 14. x

Kuta Software - Infinite Algebra 2

Properties of Parabolas Identify the vertex of each. 1) y = x 2 + 16x + 64

2) y = 2x − 4x − 2

2

3) y = −x 2 + 18x − 75

4) y = −3x 2 + 12x − 10

Graph each equation. 5) y = x 2 − 2x − 3

6) y = −x 2 − 6x − 10 y

−8

−6

−4

y

8

8

6

6

4

4

2

2

−2

2

4

6

8 x

−8

−6

−4

−2

2

−2

−2

−4

−4

−6

−6

−8

−8

4

6

8 x

6

8 x

Identify the min/max value of each. Then sketch the graph. 7) f (x) = − x 2 + 8 x − 20

8) f (x) = −

y

1 3

x2 +

4 16 x− 3 3

8

y 6

8

4

6

2

4 2

−8

−6

−4

−2

2 −2

4

6

8 x −8

−6

−4

−2

2

−4

−2

−6

−4

−8

−6 −8

4

9) f (x) = x 2 + 2x − 1

10) f (x) = −x 2 − 10 x − 30

y

−8

−6

−4

y

8

8

6

6

4

4

2

2

−2

2

4

6

8 x

−8

−6

−4

−2

2

−2

−2

−4

−4

−6

−6

−8

−8

4

Identify the vertex, axis of symmetry, and min/max value of each. 11) f (x) = 3x 2 − 54x + 241

12) f (x) = x 2 − 18x + 86

4 48 114 13) f (x) = − x 2 + x − 5 5 5

14) f (x) = −2x 2 − 20x − 46

1 15) f (x) = − x 2 + 7 4

16) f (x) = x 2 − 12x + 44

17) f (x) =

1 2 x − x+9 4

18) f (x) = x 2 + 4x + 5

6

8 x

Kuta Software - Infinite Algebra 2

Name___________________________________

Properties of Parabolas

Date________________ Period____

Identify the vertex of each. 1) y = x 2 + 16x + 64

2) y = 2x 2 − 4x − 2

(−8, 0 )

(1, −4)

3) y = −x 2 + 18x − 75

4) y = −3x 2 + 12x − 10

(9, 6 )

(2, 2)

Graph each equation. 5) y = x 2 − 2x − 3

6) y = −x 2 − 6x − 10

y

−8

−6

−4

y

8

8

6

6

4

4

2

2

−2

2

4

6

8 x

−8

−6

−4

−2

2

−2

−2

−4

−4

−6

−6

−8

−8

4

6

8 x

Identify the min/max value of each. Then sketch the graph. 7) f (x) = − x2 + 8 x − 20

8) f (x) = −

y

Max value = −4

8

1 3

x2 +

4 3

x−

16 3

y 6

8

4

6

2

4

Max value = −4

2 −8

−6

−4

−2

2 −2

4

6

8 x −8

−6

−4

−2

2

−4

−2

−6

−4

−8

−6 −8

4

6

8 x

9) f (x) = x 2 + 2x − 1

10) f (x) = −x 2 − 10 x − 30

y

−8

−6

−4

y

Min value = −2

8 6

6

4

4

2

2

−2

2

4

6

8 x

Max value = −5

8

−8

−6

−4

−2

2

−2

−2

−4

−4

−6

−6

−8

−8

4

6

8 x

Identify the vertex, axis of symmetry, and min/max value of each. 11) f (x) = 3x 2 − 54x + 241 Vertex: (9, −2 ) Axis of Sym.: x = 9 Min value = −2

4 48 114 13) f (x) = − x 2 + x − 5 5 5 Vertex: (6, 6 ) Axis of Sym.: x = 6 Max value = 6

1 15) f (x) = − x 2 + 7 4 Vertex: (0, 7 ) Axis of Sym.: x = 0 Max value = 7

17) f (x) =

1 2 x − x+9 4

Vertex: (2, 8 ) Axis of Sym.: x = 2 Min value = 8

12) f (x) = x 2 − 18x + 86 Vertex: (9, 5) Axis of Sym.: x = 9 Min value = 5

14) f (x) = −2x 2 − 20x − 46 Vertex: (−5, 4) Axis of Sym.: x = −5 Max value = 4

16) f (x) = x 2 − 12x + 44 Vertex: (6, 8) Axis of Sym.: x = 6 Min value = 8

18) f (x) = x 2 + 4x + 5 Vertex: (−2, 1) Axis of Sym.: x = −2 Min value = 1

Create your own worksheets like this one with Infinite Algebra 2. Free trial available at KutaSoftware.com

Sketching Quadratic Equations A sketch graph of a quadratic function should illustrate the following: 1. The general shape of the graph (i.e. whether there is a maximum or a minimum) with respect to the x- and y- axes. 2. The location of the y-intercept (mark on the coordinates) 3. The roots of the equation (label the location on the x-axis) 4. The location of the vertex (mark on the coordinates)

You DO NOT need to measure out an accurate scale on a sketch graph, as long as you have provided the information listed above.

Sketch graphs of the following quadratic equations, showing y-intercepts, roots, and the vertex. 2

b. y  x  12x  32

2

d. y  x  8 x 15

a. y  x 11 x 10

c. y  x  6 x  5

2

e. y  x  12 x

2

g. y   x  10 x  21

2

2

f.

y  x2  5 x

2

h. y   x  11x  10

i.

y  2 x 2  13 x  7

j.

y  2 x2  5 x  12

k.

y  x 2  4x  4

l.

y   x 2  6x  9

ANSWERS 2

2

a. shape: x y–intercept: (0, 10) Roots: (1, 0) and (10, 0) Vertex: (5.5, –20.25)

b. shape: x y–intercept: (0, 32) Roots: (4, 0) and (8, 0) Vertex: (6, – 4).

15 y 10 x 5 0 -5 0 1 2 3 4 5 6 7 8 9 10111213141516 -4-3-2-1 -10 -15 -20 -25

60 50 40 y 30 20 10 0 -2 -10 0

2

x 2

4

6

8

2

c. shape: x y–intercept: (0, 5) Roots: (– 5, 0) and (– 1, 0) Vertex: (– 3, – 4)

d. shape: x y–intercept at (0, 15) Roots at (– 3, 0) and (–5, 0) Vertex at (– 4, – 1) 20

y 10

x

0 -5

-4

-3

-2

-1

0

1

2

-8

-7

-6

-5

-4

-3

0 -1 0 -10

-2

-5

e. shape: x 2 y-intercept: (0, 0) Roots: (0, 0) and (12, 0) Vertex: (6, – 36) x

-20 -40

-6

0

1

8 6 4 2 0 -2 0 -4 -6 -8 y

0 -2

x

f. shape: x 2 y–intercept: (0, 0) Roots: – 5 and 0 Vertex: (– 2.5, – 6.25)

20 y

-4

y

10

5

-6

10

2

4

6

8

10

12

14

16

-5

-4

-3

-2

-1

x

1

12

2

2

g. shape:  x y–intercept: (0, – 21) Roots: (3, 0) and (7, 0) Vertex: (5, 4) 10

h. shape:  x y–intercept: (0, – 10) Roots: (1, 0) and (10, 0) Vertex: (5.5, 20.25). y

y

5

20

x

0 -2

-1

-5

0

1

2

3

4

5

6

7

8

9

10

10

-10

x

0

-15

-5

-20 -25

-10 0

5

10

-20

i. shape: x 2 y–intercept: (0, – 7) Roots: (– 7 ,0) and (0.5 ,0) Vertex at (– 3.25, – 28.13)

j. shape: x 2 y–intercept: (0, – 12) Roots at (– 4, 0) and (1.5, 0) Vertex at (– 1.25, – 15.13) y

y

20 15 10 5 0 -10

-9

-8

15

-7

-6

-5

-4

-3

-2

-1 -5 0

-6

x

1

-4

-2

2

12 8 4 0 -4 0 -8 -12 -16

-10 -15 -20 -25 -30

2

k. shape: x y–intercept: (0, 4) double zero and Vertex at (2, 0)

2

l. shape:  x y–intercept: (0, – 9) double zero and Vertex at (3, 0)

x

2

4...