Unit 5 Quadratics. Quadratic Functions Any function that can be written in the form.
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Transcript of Unit 5 Quadratics. Quadratic Functions Any function that can be written in the form.
Unit 5 Quadratics
Quadratic Functions
• Any function that can be written in the form
Quadratic Functions
• Graph forms a parabola
• Label the parts of the parabola
or
To find the axis of symmetry
• When
Find the vertex and los
Vertex (h,k) form of a Quadratic
• Standard Form:
Parent Function
Transformations
• You can tell what the graph of the quadratic will look like if the eq. is in (h,k) form
Sketch the graph
Sketch the graph
Sketch the graph
Sketch the graph
Sketch the graph
Identifying Important Parts on Calculator
• 2nd calc—then select max or min
Completing the Square
• Used to go from standard form to (h,k) form or to get the equation in the form of a perfect square to solve
• Steps:1. Move the constant2. Factor out the # in front of x2
3. Take ½ of middle term and square it4. Write in factored form for the perfect sq.
trinomial5. Add to both sides (multiply by # in front)6. Move constant back to get in (h,k) form
Examples
Examples
Example
Example
Example
Solving Quadratics
• You can solve by graphing, factoring, square root method, and quadratic formula
• Solutions, roots, or zeros
Solving by Graphing
1. Graph the parabola2. Look for where is
crosses the x-axis (where y=0)
3. May have 2 real, 1 real, or no real solutions
(Show on calculator)Review finding the
vertex
Solve the following by graphing
96.4
312.3
156.2
01032.1
2
2
2
2
xx
x
xx
xx
Solving Quadratics by Factoring
1. Factor the quadratic
2. Set each factor that contains a variable equal to zero and solve (zero product property)
More solving by factoring
253
168
1572
2
2
2
xx
xx
xx
You Try
Writing the Quadratic Eq.
• Write the quadratic with the given roots of ½ and -5
Write the quadratic with
• Roots of 2/3 and -2
When solving
• Graphing—not always best unless you have exact answers
• Factoring—not every polynomial can be factored
• Quadratic Formula—always works
• Square Root method—may have to complete the square first
Solving using Quadratic formula
• Must be in standard form
• Identify a, b, and c
a
acbbx
2
42
Examples
28122 xx
Examples
xx 221212
Examples
3242 2 xx
Examples
1342 xx
Discriminant
• Used to identify the “type” of solutions you will have (without having to solve)
numbertheoverradicalnoistherethatnote
acb
***
42
If the discriminant is…
• A perfect square---2 rational solutions
• A non perfect sq—2 irrational sol.
• Zero—1 rational sol.
• Negative—2 complex sol.
Identify the nature of the solution
04157.
0185.
2
2
xxex
xxex
Solving Quadratics using the Sq. Rt. method
• Useful when you have x2 = constant or a perfect sq. trinomial ex. (x-3)2=constant
1. Get the x2 by itself
2. Take the square rt. of both sides
3. Don’t forget + or – in your answer!!!
Examples
85.0164. 22 xexxex
Examples
22 216.653. xexxex
Examples
0253.0342. 22 xxexxxex
Quadratic Inequalities• Graphing quadratic inequalities in 2 variables:• Steps:• Graph the related equation• Test a point not on the graph of the parabola• Shade region that contains the point if it makes the
inequality true or shade the other region if it does not make the inequality true
• Ex. Ex. •
y x x 2 2 1 y x x 2 3 52
Graphing Quadratic Inequalities
542 xxy
Solving Quadratic Inequalities• Solving Quadratic Inequalities in one variable: May
be solved by graphing or algebraically.• To solve by graphing:• Steps:• Put the inequality in standard form• Find the zeros and sketch the graph of the related
equation• identify the x values for which the graph lies below the
x-axis if the inequality sign is < or • identify the x values for which the graph lies above the
x-axis if the inequality sign is > or
Solve by graphing
x x2 2 8 0 2 4 52x x
Solutions:_______________________ Solutions:_______________________
To solve algebraically:
• Steps:
• Solve the related equation
• Plot the zeros on a number line—decide whether or not the zeros are actually included in the solution set
• Test all regions of the number to determine other values to include in the solution set
Solve Algebraicallyx x2 3 18 x x2 11 30 0
4 12 102x x
Solving Quadratic Inequalities
2824 2 xx
Word Problems