MCF3M UNIT 3 – REPRESENT QUADRATIC FUNCTIONS
3.1 COMPLETE THE SQUARE
A process that is used to rewrite a quadratic functions from the standard form y= ax2+bx+c to the vertex form y = a(x – h)2+k.
Standard Form Vertex Form
y = ax2 + bx + c y = a(x – h)2+k
COMPLETE THE SQUARE
STEPS:
1)Factor “a” out of the first two terms.
2)Find the magic number….take ½ the b term (after factoring) and square it.
3)Add and subtract the magic number inside the bracket.
4)Remove the subtracted magic number from the bracket by multiplying it by the a.
5)Rewrite the bracket as a perfect square and collect like terms.
Examples :
- Rewrite each of the following in vertex form by completing the square.
a)y = x2 + 4x – 12
b) y = 3x2 – 6x – 4
c) y = -2x2 + 10x – 3
d)
Example: Consider the quadratic function f(x) = -3x2 + 2x – 1.
i) Write in vertex form by completing the square.
ii) State the maximum or minimum value of y, and identify which it is.
iii)State the corresponding value of x. (i.e. the axis of symmetry)
Example: A ball is thrown vertically upward from the balcony of an apartment building. The ball falls to the ground in a parabolic flight. The height of the ball, h meters, above the ground after t seconds is given by the function h = -5t2 + 15t + 35.
- Determine the maximum height of the ball.
- How long does it take the ball to reach its maximum height?
- How high is the balcony?
- Sketch the graph.
Example:
- OC Transpo carries 30000 bus riders per day for a fare of $2.25. The city of Ottawa hopes to reduce car pollution by getting more people to ride the bus, while maximizing the transit system’s revenue at the same time. A survey indicates that the number of riders will increase by 1000 for every $0.05 decrease in the fare.
a)Determine an equation to represent the revenue. Hint: Let x represent the change (increase/decrease) in ticket price. REVENUE = (TICKET PRICE) (# OF PEOPLE)
b)What fare will produce the greatest revenue?
c)What is the maximum revenue?
3.2 THE QUADRATIC FORMULA
We cannot solve every quadratic equation in standard form by factoring because many quadratic equations are not factorable. In order to solve quadratic equations that are not factorable, we can rewrite the equation in vertex form by completing the square and then isolating for x. This however seems like an awfully long process. To shorten the process, mathematicians came up with the quadratic formula.
Quadratic Formula: a formula for determining the roots of a quadratic equation in standard form ax2 + bx + c = 0.
Example: Find the roots using the quadratic formula.
a) 2x2 + 3x – 1 = 0b) 4x2 + 4x + 1 = 0
c)
Projectile Motion
Example: A digital sensor records the height of a baseball after it is hit into the air. Quadratic regression on the data gives the quadratic relation y = -4.9x2 + 20.58x + 0.491, where y is the height in metres and x is the time in seconds.
a)At what height was the ball hit?
b)After how many seconds does the ball hit the ground?
c)How long is the ball above 15m?
d)How high is the ball after 1.7 seconds?
3.3 REAL ROOTS OF QUADRATIC EQUATIONS
Solve using the quadratic formula
a) 4x2 –12x +9 = 0
b) 5x2 + 2x –2 = 0
c) x2 - x = -2
The Discriminant
b2 - 4ac is the part underneath the sign. It determines how many roots a quadratic equation will have or in how many places the parabola crosses the x-axis.
IF b2 - 4ac 0, then there are 2 different real roots.
IF b2 - 4ac = 0, then there is 1 real root OR 2 equal roots.
IF b2 - 4ac 0, then there are NO REAL ROOTS.
Examples: Without solving, determine the nature of the roots.
a) x2 – 5x + 7 = 0b) 2x2 + 4x – 3 = 0
c) x2 – 14x + 49 = 0
Finding the X-Intercepts
- To find to x-intercepts of a quadratic function in standard form, we can set y or f(x) = 0 and solve by factoring.
- OR we can set y or f(x) = 0 and solve using the quadratic formula.
Example: Find the x-intercepts of the following quadratic functions. Round your answer to the nearest hundredth, when necessary.
a) y = 2x2 – x- 4b) f(x) = 3x2 – 4x + 2
3.4 MULTIPLE FORMS OF QUARATIC FUNCTIONS
To Graph a Quadratic Function from Standard Form y = ax2 + bx + c
- Determine its key features: x-intercepts, y-intercept and vertex.
Example: The function h= -2x2 + x + 10 models the main rise and drop of a small rollercoaster, where h is the height above the ground in meters and x is the horizontal distance from a vertical support in meters.
a)Determine the x-intercepts, the h intercept and the vertex.
b)Use the information above to graph the function.
Example:
A baseball is hit off a tee. The equation h = -5t2 + 15t + 1 represents the height of the ball, in metres, t seconds after it has been hit.
a)Determine the x-intercepts, y-intercept and vertex.
b)Use the information above to graph the function.
3.5 MODEL WITH QUADRATIC EQUATIONS
Finding an Equation from a Graph
- Decide, from the information given on the graph, which form the equation will take.
- If factored form is used, substitute the x-intercepts into y=a(x–r)(x- s) for r and s.
- If vertex form is used, substitute the vertex into y= a(x – h)2+k for h and k.
- Use a point (x, y), that has not already been used and substitute it in for x and y to solve for a.
- Write your equation.
Examples: Determine the equation of the following quadratic functions.
a)
b)
Example: A water balloon is launched from a catapult at a University Science Day. The balloon reaches a maximum height of 30m and hits the ground 50m from the catapult.
a)Draw a diagram and label important points.
b)Determine an algebraic expression that models the height of the balloon at various horizontal distances.
Finding an Equation that Models a Data Set
Example: The following data was collected in an experiment that studied the flight path of a cannonball.
a)Determine an equation for the flight path.
b)Use your equation to estimate the height of the cannonball at a horizontal distance of 30m.