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Quadratic Functions

Quadratic Functions. Lesson 2.6. Applications of Parabolas. Solar rays reflect off a parabolic mirror and focus at a point This could make a good solar powered cooker. Today we look at functions which describe parabolas. Finding Zeros.

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Quadratic Functions

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  1. Quadratic Functions Lesson 2.6

  2. Applications of Parabolas • Solar rays reflect off a parabolic mirror and focus at a point • This could make a good solar powered cooker Today we look at functions which describe parabolas.

  3. Finding Zeros • Often with quadratic functions     f(x) = a*x2 + bx + c   we speak of “finding the zeros” • This means we wish to find all possible values of x for which    a*x2 + bx + c = 0

  4. Finding Zeros • Another way to say this is that we are seeking the x-axis intercepts • This is shown on the graph below • Here we see two zeros – what other possibilities exist?

  5. Factoring • Given the function   x2 - 2x - 8 = 0 •  Factor the left side of the equation    (x - 4)(x + 2) = 0 • We know that if the product of two numbers   a * b = 0     then either ... • a = 0     or • b = 0 • Thus either • x - 4 = 0    ==> x = 4     or • x + 2 = 0    ==> x = -2

  6. Warning!! • Problem ... many (most) quadratic functions are NOT easily factored!!  •  Example:

  7. The Quadratic Formula •  It is possible to create two functions on your calculator to use the quadratic formula. • quad1 (a,b,c)           which uses the    -b + ... • quad2 (a,b,c)           which uses the    -b -

  8. The Quadratic Formula • Try it for the quadratic functions • 4x2 - 7x + 3 = 0                           • 6x2 - 2x + 5 = 0 Click to view Spreadsheet Solution

  9. The Quadratic Formula • 4x2 - 7x + 3 = 0  

  10. The Quadratic Formula • Why does the second function give "non-real result?“ • 6x2 - 2x + 5 = 0

  11. Concavity and Quadratic Functions • Quadratic function graphs as a parabola • Will be either concave up • Or Concave Down

  12. Applications • Consider a ball thrown into the air • It's height (in feet) given by h(t) = 80t – 16t 2 • Evaluate and interpret h(2) • Solve the equation h(t) = 80 • Interpret the solution • Illustrate solution on a graph of h(t)

  13. Assignment • Lesson 2.6 • Page 92 • Exercises 1 – 31 Odd

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