Download MCF 3MI - U4 - 00 - All Lessons

Warm up
Solve by factoring
1) 0 = x2 + 3x - 28
2) 10x2 + x - 3 = 0
Solve by rearranging to isolate x
3) 2(x - 3)2 - 8 = 0
The Quadratic Formula
Quadratic equations are of the form____________________.
The roots of the quadratic equation correspond to____________________________________.
We have solved quadratic equations by


If we cannot solve 0 = ax2 + bx + c by factoring, we can use…
The Quadratic Formula
Example: Solve using the quadratic formula. List a, b & c before substituting into formula.
x2 - 5x - 6 = 0
Solve using the quadratic formula
1) 2x2 - 7x = - 3
3) 3x2 + 4x + 8 = 0
2) x2 + 5x - 4 = 0
Warm up
1) Solve by factoring
0 = x2 - 2x - 15
2) Solve using the quadratic formula
0 = x2 - 2x - 15
Quadratic Formula II
All quadratic equations of the form ax2 + bx + c = 0 can be solved using the quadratic formula:
x
b  b2  4ac
2a
1) Solve. Verify your answer with a graphing calculator.
2x2 + 6x – 3 = 0
2) Solve.
–n2 + 5 = 0
3) Solve. Round your answer to the nearest hundredth.
2.5x2 + 14.2x - 5.8 = 0
Warm up
Solve using the quadratic formula.
(Determine the roots of each equation)
1) x2 - 10x + 25 = 0
2) 4x2 + 4x +1 =0
3) 3x2 - x + 5 = 0
Nature of the Roots
How many x-intercepts (zeros) can a quadratic function have?
How many solutions can a quadratic equation have?
Example: Determine the number of roots for each of the following equations.
1) x2 - 5x -11 = 0
2) 3x2 - 4x = -6
3) 9x2 - 60x + 100 = 0
4) 2(x - 1)2 + 3 = 0
Example:
For what values of k does the equation 2x2 - 8x + k = 0 have 2 roots? 1 root? no roots?
Warm up
The graph of f(x) = 0.5(x - 5)2 + 3 is shown below. Complete the information.
vertex:
axis of symmetry:
direction of opening:
max or min value:
domain:
range:
x-intercept(s):
y-intercept:
state f(3):
Quadratic Functions
Vertex and Standard Forms
Vertex Form
Standard Form
Example 1: Complete the table
f(x) = (x - 2)2 - 4
vertex
axis of symmetry
direction of opening
max/min value
domain & range
g(x) = -3(x + 1)2 +5
Example 2: Write the equation of each function in vertex and standard form.
i)
ii) vertex (-1, 3) passing through (1, -9)
Example 3:
A water balloon is thrown from the balcony of an apartment building, following a path defined by
h(t) = -5(t - 2)2 + 32 , where h is in meters and t is in seconds.
1) What is its maximum height?
2) When does it reach its maximum height?
3) Write the equation in standard form.
4) What height was it thrown from?
Warm up
Graph y = 2(x – 3)2 - 4
vertex:
axis of symmetry:
max or min:
domain:
range:
Standard and Vertex Form
vertex form:
standard form:
factored form:
Example 1: Covert to vertex form by "completing the square."
f(x) = 2x2 + 12x + 11
Step 1: Factor "a" out of the first two terms
Step 2: Add and subtract the special number
Step 3: Pull the special number out of the brackets
Step 4: Factor the perfect square trinomial
State the vertex, axis of symmetry, max or min value, domain and range of f(x).
Example 2: Covert to vertex form by "completing the square."
g(x) = x2 + 8x + 7
Example 3: Covert to vertex form by "completing the square."
h(x) = 5x - 0.5x2
Warm up
Find the vertex by completing the square.
f(x) = 2x2 + 20x – 13
Step 1: Factor "a" out of the first two terms
Step 2: Add and subtract the special number
Step 3: Pull the special number out of the brackets
Step 4: Factor the perfect square trinomial
Also state the axis of symmetry, the max or min value, the domain and the range.
Completing The Square
Example 1: Write in vertex form. State the minimum value of the function and when it occurs.
y = x2 + 5x - 6
Example 2: Write in vertex form. State the maximum value of the function and when it occurs.
How is the quadratic formula related to completing the square?
p. 218 in your textbook
ax2 + bx + c = 0
ANSWERS
1) min of -43 at x = -6
4) max of 44 at x = -7
7) min of -1 at x = 2
10) min of -5/4 at x = -3/2
12) min of -1/3 at x = -1/3
2) max of 10 at x = 3
5) max of 35 at x = -5
8) max of 3 at x = -1
11) min of -9/4 at x = 1/2
13) max of -8 at x = 1/2
3) min of -87 at x = 10
6) min of -18 at x = -3
9) min of -59 at x = 3
Warm up
Consider the function y = -3x2 + 6x + 9
1) Convert it to vertex form
2) Convert it to factored form
3) State all of the important
points/features of this
parabola.
More Completing the Square
Example 1: The flight of a projectile is modeled by the equation h(t) = -5t2 + 30t + 4
a) What is the maximum height reached by the projectile?
b) When does it reach the maximum height?
c) What was the initial height?
d) When does it hit the ground?
Example 2: The sum of two number is 32. Find the maximum product of the numbers.
Practice Questions:
p. 215 #10, 11
1. a) Find the minimum product of two numbers whose difference is 12.
b) What are the two numbers?
2. a) Find the maximum product of two numbers whose sum is 23.
b) What are the two numbers?
3) A rectangle has dimensions 3x and 5 - 2x.
a) What is the maximum area of the rectangle?
b) What value of x gives the maximum area?
4. The path of a thrown baseball can be modelled by the function h = -0.004d2 + 0.1d + 2 where h is the
height of the ball, in metres, and d is the horizontal distance of the ball from the player, in metres.
a) What is the maximum height reached by the ball?
b) What is the horizontal distance of the ball from the player when it reaches its maximum height?
c) How far from the ground is the ball when the player releases it?
ANSWERS
1a) -36
b) -6 and 6
2 a) 132.25
b) 11.5 and 11.5
3 a) 9.375 square units
b) 1.25
4 a) 3.23 m
b) 17.5 m
c) 2 m
Solving Problems involving Quadratic Models
There are essentially three types of problems if our relation is quadratic
1)
Given x, find the value of y
2)
Given y, find the value of x
3)
Find the maximum or minimum value of y
Example
A sporting goods store sells 90 ski jackets per season for $200 each. Each $10 decrease in
price would result in 5 more jackets being sold.
Find
 the price that yields maximum revenue.
 maximum possible revenue.
 the number of jackets sold & price if revenue is $17 600.
Warm up
The height of a rock launched from a slingshot is modelled by the function h(t) = -5t2 + 20t + 2
where h is the height in metres and t is the time in seconds.
a) What height was the rock launched from?
b) When does the rock hit the ground?
c) What is the maximum height reached by the rock?
More Quadratic Models
Example from Yesterday
A sporting goods store sells 90 ski jackets per season for $200 each. Each $10 decrease in
price would result in 5 more jackets being sold.
Find
1) the price that yields maximum revenue.
2) maximum possible revenue.
3) the number of jackets sold & price if revenue is $17 600.
Alternate methods...
R(x) = (200 - 10x)(90 + 5x)
= -50x2 + 100x + 18000
Warm up
Determine the equation of the function in vertex form that has vertex (3,4) and passes through
the point (-1,6).
Quadratic Functions
The motion of a ball rolled up a ramp is shown in the chart below. Time (t) is in seconds and the
height (h(t)) is in meters.
 Create a scatterplot.
 Draw a line/curve of best fit.
 Identify the type of Function.
 Find the equation of the function.
 Check with a graphing calculator.
 Use your equation to find the height of the ball after 2.25 seconds.