Quadratics: standard form
by: kunal chopra
- 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"
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
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.
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
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)
GRAPHING STANDARD FORM
- 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
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
- 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
- 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.
y = 2x² + 2x - 3
D = b² - 4ac
= (2)² - 4(2)(-3)
= 4 + 24
Therefore, since 28 is greater than O, this quadratic will have two x-intercepts.
a) Write an equation to represent Sherri's total sales revenue.
[Hint: R = (price)(quantity)]
Let x represent the number of price decreases.
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
Therefore, to maximize her revenue, Sherri must charge $8.75 per photo.
- 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
- 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
- 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
- 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