Quadratics: Standard Form

By: Shawn Singh

Success Criteria: Learning Goals


  • To Complete the Square in order to find the maximum/minimum value of the quadratic and its vertex
  • To use the Quadratic Formula in order to find the x-intercepts along with the vertex of a quadratic function
  • To find how much x-intercepts a quadratic function has by finding the Discriminant
  • To graph a quadratic function using the "Quadratic Formula"

Summary

  • Standard Form Equation: y = ax² + bx + c
  • a value gives you the shape and direction of opening of the quadratic
  • c value gives you the y-intercept of the quadratic
  • To get x-intercepts, SOLVE using the "Quadratic Formula"
  • Quadratic Formula: x =[ -b + √(b² - 4ac)] / 2a *on picture*
  • To identify if the quadratic has a maximum or minimum value, complete the square to transform standard form equation in vertex form
  • Vertex Form Equation: y = a(x - h)² + k
  • To identify how much x-intercepts a quadratic has, use the discriminant formula
  • Discriminant Formula : x = b² - 4ac

Completing the Square

  • Involves transforming a standard form equation into vertex form in order to get the vertex and maximum or minimum value of the quadratic
Below are steps in order of how to change a standard form equation into vertex form


Ex. 1

y = x² - 8x + 10

y = (x² - 8x )+ 10 Step One: Isolate ax² and bx terms, also (-8/2)² basically (b/2)²

y = (x² - 8x + 16 - 16 ) + 10 Step Two: Add and subtract the value from above

y = (x² - 8x + 16) + 10 - 16 Step 3: Bring the -16 outside the bracket

y = (√ x² - 8x + √ 16) - 6 Step 4: Square root both ax² and c terms)

y = (x - 4)² Step Five: Write as a perfect square.


Ex. 2

y = 2x² + 20x - 8

y = 2(x² + 10x) - 8 *If there is a coefficient, factor it out (applies to ax² and bx)

y = 2(x² + 10x + 25 - 25) - 8 *Remember, multiply coefficient with subtracted value (2)(-25)

y = 2(√ x² + 10x + √ 25) - 50 - 8

y = 2(x + 5)² - 58


Ex. 3

y = -x² + 10x + 20

y = -(x² - 10x) + 20 *Even if there is a negative sign, factor it out*

y = -(x² - 10x +25 - 25) + 20

y = -(√ x² - 10x + √ 25) + 20 + 25

y = -(x - 5)² + 45


*The video below will help further your understanding of Completing the Square*

❖ Completing the Square - Solving Quadratic Equations ❖

The Quadratic Formula

  • Using the quadratic formula is a direct way of calculating roots (also known as zeroes, the solution or the x-intercepts)
  • A quadratic equation can have 2, 1 or 0 solutions
  • If the number under the square root is negative, the quadratic has no solution
  • Quadratic Formula: x =[ -b + √(b² - 4ac)] / 2a


Ex. 1

9x² - 24x + 16 = 0

x =[ -b + √(b² - 4ac)] / 2a

x =[ -(-24) + √((-24)² - 4(9)(16))] / 2(9)

x =[ 24 + √(576 - 576)] / 18

x =[ 24 + √0)] / 18


x = (24 + 0) / 18

x = 24 / 18

x = 1.3 or 4/3

(1.3, 0) or (4/3, 0)


Ex. 2

2x² + 4x = -1

2x² + 4x - 1 = 0

x =[ -b + √(b² - 4ac)] / 2a

x =[ -4 + √((4)² - 4(2)(1))] / 2(2)

x =[ -4 + √(16 - 8)] / 4

x =[ -4 + √8] / 4


x = (-4 + 2.8) / 4

x = -1.2/4

x = -0.3


x = (-4 - 2.8) / 4

x = -6.8/4

x = -1.7

(-0.3, 0) (-1.7, 0)


*MY OWN VIDEO BELOW will help further your understanding of using the Quadratic Formula*

The Quadratic Formula

Graphing Standard Form

In order to graph a standard form equation follow these steps:
  • Step One: Find the x-intercepts using "The Quadratic Formula"
  • Step 2: Find the vertex (Use AOS = (r + s)/2 to find x-value and sub in x-value into original equation to find y-value of vertex
  • Plot the vertex along with the two x-intercepts to form parabola


Ex. 1

y = x² + 2x - 3

Step One: x-intercepts

x =[ -b + √(b² - 4ac)] / 2a

x =[ -2 + √((2)² - 4(1)(-3))] / 2(1)

x =[ -2 + √(4 + 12)] / 2

x =[ -2 + √(16)] / 2


x = (-2 + 4) / 2

x = 2/2

x = 1


x = (-2 - 4) / 2

x = -6/2

x = -3

x- intercepts = (1, 0) (-3, 0)


Step Two: Vertex

AOS = (r + s)/2 = (-3 + 1) / = -2/2 = -1

y = x² + 2x - 3

y = (-1)² + 2(-1) - 3

y = 1 - 2 - 3

y = - 4

Vertex (-1, -4)


Step 3: Plots x-Intercepts and Vertex on Graph *graph on side*

The Discriminant

  • The Discriminant Formula is used to identify the number of x-intercepts in a quadratic function
  • Given that a, b and c are rational numbers, they are needed to determine the Discriminant value.
  • Discriminant Formula = b² - 4ac
The following will tell you how to determine how much x - intercepts a quadratic function has:


  • D>0 = 2 Solutions - If D is greater than 0 the quadratic will have two x-intercepts.
  • D<0 = 0 Solutions - If D is less than 0 the quadratic will have no x-intercepts.
  • D = 0 = 1 Solution - If D is equal to 0, the quadratic will have 1 x-intercept.


Ex. 1

y = 2x² + 2x - 3

D = b² - 4ac

= (2)² - 4(2)(-3)

= 4 + 24

= 28

Therefore, since 28 is greater than O, this quadratic will have two x-intercepts.


Ex. 2

y = x² - 2x + 5

D = b² - 4ac

= (-2)² - 4(1)(5)

= 4-20

= -16

Therefore, since -16 is less than 0, this means the quadratic will have no x-intercepts.


Ex. 3

y = x² + 2x + 1

D = b² - 4ac

= (2)² - 4(1)(1)

= 4 - 4

= 0

Therefore, since D = 0, this means the quadratic will have only 1 x-intercept.

Word Problem #1 - Complete the Square

Sherri sells photos of athletes to baseball, basketball and hockey fans after their games. Her regular price is $10 per photo, and she usually sells about 30 photographs. Sherri finds that, for each $0.50 reduction in price, she can sell two more photographs.


a) Write an equation to represent Sherri's total sales revenue.

[Hint: R = (price)(quantity)]

Let x represent the number of price decreases.

EQUATION:

R = (10 - 0.50x)(30 + 2x)


b) What price must Sherri charge in order to maximize her revenue.

R = (10 - 0.50x)(30 + 2x)

= 300 + 20x - 15x - x²

= - x² + 5x + 300

= - (x² - 5x) + 300

= - (x² - 5x + 6.25 - 6.25) + 300

= - (x² - 5x + 6.25) + 300 + 6.25

= - (√x² - 5x + √6.25) + 306.25

= - (√x² - 2.5)² + 306.25

Vertex (2.5, 206.25)

Price = 10 - 0.50x

= 10 - 0.50(2.5)

= 10 - 1.25

= 8.75

Therefore, to maximize her revenue, Sherri must charge $8.75 per photo.

Word Problem #2 - The Quadratic Formula

The product of two consecutive numbers is 3306. What are the numbers?


Let x represent the first number.

Let x + 1 represent the second number.


x(x + 1) = 3306

x² + x = 3306

x² + x - 3306 = 0

x = [-b + √(b² - 4ac)] / 2a

x = [-1 + √((1)² - 4(1)(- 3306))] / 2(1)

x = [-1 + √( 1 + 13224)] / 2

x = [-1 + √13225] / 2


x = (-1 +115) / 2

x = 114 / 2

x = 57


x = (-1 -115) / 2

x = -116 / 2

x = -58


x = 57

x + 1 = 57 + 1 = 58

Check: (57)(58) = 3306


x = -58

x + 1 = -58 + 1 = -57

Check: (-58)(-57) = 3306


Therefore, the numbers are 57 and 58 or -57 and -58.

Reflection


  • Overall the Quadratics Unit has been a fairly interesting unit to learn about where it connects to a whole lot of applications throughout the world
  • With the exception of a few silly mistakes, I was able to exceed throughout this whole unit in all the 3 components and understand as well as apply the key concepts of the quadratics unit



Connections:

1. Vertex Form Connects to Standard Form
  • Standard Form can be obtained by expanding Vertex Form
  • y = a(x – h)² + k -------> y = ax² + bx + c = 0
  • Vertex Form can be obtained by "Completing the Square" from Standard Form
  • y = ax² + bx + c = 0 -------> y = a(x – h)² + k
2. Vertex Form Connects to Graphing
  • Once your quadratic function is put into vertex form you are able to identify the vertex of the function and plot it on your graph
  • Once vertex is plotted, use the step pattern (a value included) to continue graphing from both sides

3. Standard Form Connects to Graphing

  • In a standard form equation, the c value gives you the y-intercept, begin with plotting the y-intercept on the graph
  • Transforming your Standard Form Equation into Vertex Form by Completing the Square, will give you your vertex as well as your a value
  • Once vertex is plotted use the a value as well as the step pattern to continue to plot your quadratic

4. Factored Form Connects to Graphing

  • In a factored form equation the two x-intercepts as well your a value is easily obtainable by factoring your equation. Plot your two x-intercepts
  • Use x-intercepts to find the vertex of the equation (x-value - AOS, y - value subbing x into original equation
  • Once vertex is plotted use a value along with the step pattern to graph quadratic

5. Factored Form Connects to Standard Form

  • In order to find the vertex, this two forms involve similar methods involving finding the vertex
  • In Factored Form: Once two x-intercepts are found by factoring, find the vertex by using AOS formula (x), and subbing x into the original equation for (y).
  • In Standard Form: Once two x-intercepts are found by using the "Quadratic Formula", find the vertex by using AOS formula (x), and subbing x into the original equation for (y).

6. All Forms Connect to Each Other (Vertex, Factored, Standard)

  • In order to find the y-intercept for all of these forms, when x is subbed in for 0 (x = 0) the y-intercept is easily obtainable in all forms
  • The y intercept is written as (0, y) - for all forms

Assessment

  • This assessment strengthened my knowledge to understand that their are multiple ways to come to a solution
  • In terms of finding the vertex, I originally did it by using the Quadratic Formula to find the x-intercepts then went along to find the vertex
  • I ensured that I was right by Completing the Square to the the equation in Vertex Form to obtain the vertex of this quadratic relation
  • This made me realize there's more than one way of doing a problem involving quadratics and doing it a second time and arriving at the same answer will ensure that your answer is correct
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Quadratics: Standard Form

Leaning Quadratics in Standard Form!

One Step at a Time!