graphing quadratic functions in standard form...

FOM 11 T28 – GRAPHING QUADRATIC FUNCTIONS IN STANDARD FORM 1 1

© R. Ashby 2018. Duplication by permission only.

GRAPHING QUADRATIC FUNCTIONS IN STANDARD FORM I) THE STANDARD FORM OF A QUADRATIC FUNCTION (PARABOLA) IS y = ax 2 +bx +c . To graph a quadratic

function in standard form you must know where the vertex is relative to the y-axis. The vertex may be to the left of, on or to the right of the y-axis. Recall that c indicates the location of the y-intercept. We can use the signs of a and p to indicate the position of the vertex relative to the y-axis. Complete the investigation below to learn how to determine the position of the vertex relative to the y-axis of a quadratic function in standard form.

II) INVESTIGATION 1: Answer the questions for each quadratic function. Answer the summary questions at the end of the investigation.

1) y = x2

Write the function in perfect standard from:

List a = b = c =

Sign of a = Value of b =

State the position of the vertex relative to the y-axis.

2) y = x2 + 4


List a = b = c =



3) y = x2−5


List a = b = c =



x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-6 -4 -2 0 2 4 6

2

4

6

8

10

12

14

16

18

x

y

-6 -4 -2 0 2 4 6

-6

-4

-2

2

4

6

8

10

12

14



4) y =−2x2 +8


List a = b = c =



5) y = x2 + x


List a = b = c =

Sign of a = Sign of b =


6) y = x2 + 2x


List a = b = c =



x

y

-5 -4 -3 -2 -1 0 1 2 3 4

-2

2

4

6

8

10

12

14

x

y

-6 -5 -4 -3 -2 -1 0 1 2 3 4

-2

2

4

6

8

10

12

14



7) y = 2x2 + 3x


List a = b = c =



8) y = −x2− x


List a = b = c =



9) y = −x2−2x


List a = b = c =



x

y

-4 -3 -2 -1 0 1 2 3

-2

2

4

6

8

10

12

14

x

y

-5 -4 -3 -2 -1 0 1 2 3 4

-14

-12

-10

-8

-6

-4

-2

2

x

y

-5 -4 -3 -2 -1 0 1 2 3

-14

-12

-10

-8

-6

-4

-2

2



10) y = −x2 + x


List a = b = c =



11) y = −x2 + 2x


List a = b = c =



12) y = −2x2 + 3x


List a = b = c =



x

y

-3 -2 -1 0 1 2 3 4

-14

-12

-10

-8

-6

-4

-2

2

x

y

-3 -2 -1 0 1 2 3 4

-14

-12

-10

-8

-6

-4

-2

2

x

y

-3 -2 -1 0 1 2 3 4

-14

-12

-10

-8

-6

-4

-2

2



13) y = x2− x


List a = b = c =



14) y = x2−2x


List a = b = c =



Conclusions:

The vertex is on the y-axis when b = .

The vertex is on the left side of the y-axis when the signs of a & b are the .

The vertex is on the right side of the y-axis when the signs of a & b are .

III) Steps to Graph Quadratic Functions in Standard Form

A) USE THESE STEPS TO SOLVE OPTIMIZATION PROBLEMS

1: Use the signs of a & b to determine the position of the vertex relative to the y-axis. 2: Create a table of ordered pairs using values of x starting at x = 0 for the side of the x-axis that where the

vertex is located until the vertex is located. 3: Plot the points on a grid. Use the axis of symmetry to plots corresponding points on the other side of the

parabola.

x

y

-2 -1 0 1 2 3

-14

-12

-10

-8

-6

-4

-2

2

4

x

y

-2 -1 0 1 2 3

-2

2

4

6

8



B) SAMPLE PROBLEMS 1: Study these examples carefully. Be sure you understand and memorize the process used to complete them. Instructions: Graph these quadratic functions. 1. y = x2−7

1: Use the signs of a & b to determine the position of the vertex relative to the y-axis.

a = & b =

∴ The vertex is the y-axis

2: Create a table of ordered pairs using values of x starting at x = 0 for the side of the x-axis that where the vertex is located until the vertex is located.

b = , therefore the vertex is on the y-axis. Use x = 0, 1, 2 & 3 to determine ordered pairs.

x y x2 – 7

3: Plot the points on a grid. Use the axis of symmetry to plots corresponding points on the other side of the parabola.

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10



2. y = x2−4x 1: Use the signs of a & b to determine the position of the vertex relative to the y-axis.

a = & b =

∴ The vertex is on the side of the y-axis


a & b have signs, therefore the vertex is on the side of the y-axis. Use x = , etc. to determine ordered pairs.

x y x2 – 4x

3: Plot the points on a grid. Use the axis of symmetry to plots corresponding points on the other side of

the parabola.

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10



3. y = x2 + x−2 1: Use the signs of a & b to determine the position of the vertex relative to the y-axis.

a = & b =

∴ The vertex is on the side of the y-axis


a & b have the signs, therefore the vertex is on the side of the y-axis. Use x = , etc. to determine ordered pairs.

x y x2 + x – 2

In this case we must calculate the coordinates of the vertex as illustrated here.

Calculate the equation of the axis of symmetry:

Calculate the minimum:

State the coordinates of the vertex:

3: Plot the points on a grid. Use the axis of symmetry to plots corresponding points on the other side of the parabola.

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10



C) REQUIRED PRACTICE 1: Complete these questions. 1) Draw the graphs for the quadratic functions in question 5 on page 370. 2) Answer question 11 on page 370. 3) Draw the graph for the quadratic function in question 2a on page 369.

DO NOT USE THESE GRIDS FOR YOUR ASSIGNMENT

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10



x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

graphing quadratic functions in standard form...

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