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Determining Maximum and Minimum Values of a quadratic Function!! - PowerPoint PPT Presentation


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Determining Maximum and Minimum Values of a quadratic Function!!. Sometimes there is a little confusion…. When a question asks for the maximum or the minimum of a quadratic function, it is not asking for the whole vertex. It is simply asking for the y coordinate of the vertex

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sometimes there is a little confusion
Sometimes there is a little confusion…..
  • When a question asks for the maximum or the minimum of a quadratic function, it is not asking for the whole vertex.
  • It is simply asking for the y coordinate of the vertex
  • It’s important to keep this in mind because when answering word problems etc…..we don’t want to do more work than is necessary.
what equations tell us
What equations tell us.
  • By looking at an equation in standard form, factored form or vertex form, we can immediately tell whether the parabola will have a max or min.
how is this you ask
How is this, you ask?
  • Simply look at the “a” value.
  • If a > 0 the parabola opens up and we have a minimum value
  • If a < 0 the parabola opens down and we have a maximum value.
finding the max or min value there are lots of ways
Finding the max or min value….there are lots of ways.
  • If the equation is in standard form we can:
    • Complete the square to put it in vertex form
    • Express the equation in factored form, find the midpoint of the zeros to get the axis of symmetry etc
    • Use quad formula to find the zeros
    • Use a graphing calculator (we won’t discuss this here
let s start out with completing the square
Let’s start out with completing the square.
  • Unfortunately…..there are some things in life that you just have to remember…..
  • Procedures, procedures, procedures
  • Have you ever watched a pilot before a flight?
  • What does he/she do?
  • Why?
  • After lots of practice does he/she have it pretty well memorized…..you don’t have to think about it anymore hopefully…
procedures
Procedures….
  • Completing the square algebraically is much like this.
  • You have to follow a set of rules to get you to your desired destination.
  • Follow along and fill out your spread sheet as we complete the square for the equation y=x2+12x+40
step 1
Step 1

Insert brackets around the first two terms

step 2
Step 2

Factor out any value in front of the x2

In this case there is nothing to factor out

step 3
Step 3

Take Middle term divide by 2 & square it

step 4
Step 4

Rewrite the equation with the result from step 3 added and subtracted inside the brackets

step 5
Step 5

Bring the (-) term outside of the bracket remember to remultiply if necessary

step 6
Step 6

Combine the two constant terms outside the bracket

step 7
Step 7

Factor trinomial inside the bracket

step 8
Step 8

Express answer in vertex form

and that s all folks
And that’s all folks……

There isn't much more to this procedure.

Of course there are more difficult equations

But the process essentially remains the same

recall
Recall….
  • The factored form of a quadratic equation is
when an equation is in factored form
When an equation is in factored form….
  • The axis of symmetry is the vertical line that runs through the midpoint of the x intercepts.
  • This can be found by the following calculation
  • This value is also the x coordinate of the vertex.
when an equation is in factored form1
When an equation is in factored form
  • Recall, you find the y coordinate of the vertex by plugging the x coordinate of vertex into original equation and solving for y.
  • So Remember……..
  • The x value of the vertex gives the axis of symmetry,
  • The ‘y’ value gives the max/min value of the function
example
EXAMPLE

Find the zeroes, vertex, AOS, max/min for the function

y=3(x-8)(x+5)

  • X coordinate of vertex
  • Y coordinate of vertex
the quadratic formula
The Quadratic Formula
  • Unfortunately we can’t always solve quadratic equations by factoring.
  • As a result , a formula has been developed that will always let us solve for the x intercepts
the quadratic formula1
The Quadratic Formula
  • If f(x) = ax2 + bx + c is given then we could use the quadratic formula to find the roots of the equation.
  • The quadratic formula:
  • The “x” represents the x intercept.
  • The “+/-” represents the fact that there could be two x intercepts
  • a is the coefficient in front of the x2 term
  • b is the coefficient in front of the x term
  • c is the number/ constant term at the end
example solve for the vertex
Example solve for the vertex.
  • Using the Quadratic Formula solve 2x2 - 5x – 1 = 0

a = 2

b = -5

c= -1

Therefore, there are two solutions

x = 2.69 and x = -0.19

slide27
Next….
  • Plug this into original equation.
once more thing
Once more thing
  • If your equation is in standard form there is a short cut to getting the x coordinate of the vertex.
for example
For example:
  • Find the minimum of
  • Now plug this into the original equ:

The minimum is –0.25 and it occurs where x = -3.5