Quadratic Functions

​Quadratic Function

​Objectives:

• Define and illustarte quadratic equation and quadratic function.

 

• Represent a quadratic function using: (a) table of values; (b) graph; and (c) equation

​Brain Break

​https://www.youtube.com/watch?v=a9maEYcehyE

​Quadratic Equation and Quadratic Function

​Example 1: Determine whether the given equation is a quadratic function or not.

​Example 1: Determine whether the given equation is a quadratic function or not.

Multiple Choice

True or False.

4x² - 3x + 5 = y is a univariate quadratic equation.

• True

• False

Multiple Choice

Which of the following is a quadratic function?

• y = 3y² - 8

• f(x) = 9²

• h(x) = 3 + 2x² + x

• $$\frac{1}{x}=\ \frac{2x}{5}+1$$  

A table of values can be generated from a quadratic function by substituting the x-values and calculating the values for f(x).

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How can the solution(s) to a quadratic equation be found from a table of values?​

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When looking at a table of values for a quadratic function, the x-intercepts represent the x-values where y = 0. This corresponds to the x-values where f(x) is 0 in function notation.

​Representation of Quadratic Functions

A. Table of Values​

​Representation of Quadratic Functions

A. Table of Values​

​Example 2: Given the quadratic functions (a) y = x² - 1 and

(b) f(x) = x² - 3x + 2, complete the table of values below.

​y = x² - 1

​f(x) = x² - 3x + 2

​Representation of Quadratic Functions

A. Table of Values​

​Example 2: Given the quadratic functions (a) y = x² - 1 and

(b) f(x) = x² - 3x + 2, complete the table of values below.

​y = x² - 1

​f(x) = x² - 3x + 2

​zero/x-intercept

​zero/x-intercept

​zero/x-intercept

​zero/x-intercept

Multiple Choice

Given the table of values of the quadratic function at the left, what is the missing value?

• 40

• 25

• 20

• 0

Multiple Choice

Given the table of values of quadratic function at the left, what are the zeros of function?

• 0 & 1

• 1 & -3

• 1 & - 5

• -5 & - 3

​Representation of Quadratic Functions

B. Graph​

​Representation of Quadratic Functions

B. Graph​

​Representation of Quadratic Functions

B. Graph​

​Translation

​-is the horizontal and vertical movement of the graph of f(x).

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Transformation of f(x) = x² + k (Vertical Shifts)

​If k is positive, the graph of y = x² is shifted upwards by k units.

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If k is negative, the graph of y = x² is shifted downwards by k units.

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​Representation of Quadratic Functions

B. Graph​

​Transformation of f(x) = (x - h)² (Horizontal Shifts)

​If h is positive, the graph of y = x² is shifted h units to the right.

​

If h is negative, the graph of y = x² is shifted h units to the left.

​

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​Representation of Quadratic Functions

B. Graph​

​Transformation of f(x) = a(x - h)² + k with vertex (h, k)

​The axis of symmetry is x = h.

​If a > 0, the parabola opens upward and the minimum function value is k.

​

​If a < 0, the parabola opens downward and the maximum function value is k.

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​Representation of Quadratic Functions

B. Graph​

​Example 3: Given the graphs of quadratic functions below, determine the following:

​a) vertex

b) axis of symmetry

c) x-intercept/s

d) y-intercept​

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​Representation of Quadratic Functions

B. Graph​

​Example 3: Given the graphs of quadratic functions below, determine the following:

​Representation of Quadratic Functions

B. Graph​

​Example 4: Given the graphs of quadratic functions below, determine the following:

​a) vertex

b) axis of symmetry

c) x-intercept/s

d) y-intercept

e) function​

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​Representation of Quadratic Functions

B. Graph​

​Example 4: Given the graphs of quadratic functions below, determine the following:

​Representation of Quadratic Functions

B. Graph​

​Example 5: Given the graphs of quadratic functions below, determine the following:

​a) vertex

b) axis of symmetry

c) x-intercept/s

d) y-intercept

e) function ​

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​Representation of Quadratic Functions

B. Graph​

​Example 5: Given the graphs of quadratic functions below, determine the following:

​Representation of Quadratic Functions

B. Graph​

​Example 6: Compare the graph of the first equation to the graph of the second equation.

​a) y = 3x², y = x²

​Both graphs open upwards. The graph of y = 3x² is narrower than the graph of y = x².

​Representation of Quadratic Functions

B. Graph​

​Example 6: Compare the graph of the first equation to the graph of the second equation.

​c) y = (x - 3)² + 9, y = x² + 9

​d) y = (x - 5)² - 4, y = x²

​Both graphs open upwards. The value of h in

y = (x - 3)² + 9 is 3. Hence, its graph is translated 3 units to the right of the graph of

y = x² + 9.

​Both graphs open upwards.

k = -4 in y = (x - 5)² - 4 while

h = 5 in y = (x - 5)² - 4. Hence, the graph of y = (x - 5)² - 4 is translated 4 units downward and 5 units to the right of the graph of y = x².

Multiple Choice

What is the vertex of the function?

• (1, 3)

• (3, 1)

• (2, 4)

• (4, 2)

Multiple Choice

Which of the following describes the function graph?

• y = (x - 1)² + 3

• y = -(x + 1)² + 3

• y = -(x - 1)² + 3

• y = (x - 1)² - 3

Multiple Choice

What are the x-intercepts of the function?

• 1 and -1

• -1 and 0

• -2 and 0

• 2 and 0

​Representation of Quadratic Functions

C. Equation​

​Forms of Quadratic Functions

​

​General Form :

​f(x) = a(x - h)² + k

​Vertex Form :

​a :

​(h, k) :

​x = h :

​leading coefficient

​vertex

​axis of symmetry

​f(x) = ax² + bx + c

​Representation of Quadratic Functions

C. Equation​

​Vertex of a Parabola

​axis of symmetry :

​x-coordinate of the vertex:

​y-coordinate of the vertex:

​Representation of Quadratic Functions

C. Equation​

​Example 7: Write y = (x + 3)² - 5 in quadratic form.

​Solution:

​y = (x + 3)² - 5

​y = (x + 3)(x + 3) - 5

​y = x² + 3x + 3x + 9 - 5

​y = x² + 6x + 4

​Representation of Quadratic Functions

C. Equation​

​Example 8: Write y = x² - 4x - 5 in vertex form. Identify the vertex and the axis of symmetry.

​Solution:

​y = x² - 4x - 5 use completing the square

​y = (x² - 4x + 4) - 4 - 5

​y = (x² - 4x + 4) - 9

​y = (x - 2)² - 9

​vertex: (2, -9)

​axis of symmetry: x = 2

​Representation of Quadratic Functions

C. Equation​

​Example 9: Identify the vertex and the axis of symmetry of

y = -x² + 2x + 2.

​Solution:

​a = -1, b = 2, c = 2

​axis of symmetry: x = 1

​Representation of Quadratic Functions

C. Equation​

​Example 9: Identify the vertex and the axis of symmetry of

y = -x² + 2x + 2.

​Alternative solution:

​a = -1, b = 2, c = 2

​axis of symmetry: x = 1

Multiple Choice

Identify the axis of symmetry of

y = x² - 4x - 8

• 1

• 2

• -4

• 8

Multiple Choice

Identify the vertex of

y = x² - 4x - 8

• (-2, 4)

• (4, 8)

• (2, -12)

• 8, -12)

Multiple Choice

Which of the following is the general form of y = (x - 1)² + 3?

• y = x² - 2x + 4

• y = x² + 2x + 2

• y = x² - 2x - 4

• y = x² - 2x - 2

Multiple Choice

Which of the following is the vertex form of y = x² - 6x + 8?

• y = (x 3)² - 17

• y = (x - 3)² + 1

• y = (x + 3)² - 17

• y = (x - 3)² - 1

Worktext Reference:

Oronce O.A., Mendoza M.O. (2019). E-Math Worktext in Mathematics 9, Manila, Philippines: REX Bookstore Inc.

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Reference:

Nivera G.C., Lapinid M.R.C. (2018). Grade 9 Mathematics Patterns and Practicalities 9, Makati City, Philippines: Salesiana Books by Don Bosco Press, Inc.

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Albay E.M., Oli M.C. (2018). Practical Math 9, Makati City, Philippines: Diwa Learning Systems Inc.

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Web Reference:

https://www.texasgateway.org/resource/solving-quadratic-equations-using-tables#section-id-138421​

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Digital Graphing Tool:

https://www.geogebra.org/graphing?lang=en 

https://www.desmos.com/calculator​