Transcript Chapter 9 Quadratic Equations and Functions

Chapter 9
Quadratic Equations
and Functions
By: Courtney Popp & Kayla Kloss
9.1-Solving Quadratic Equations by
Finding Square Roots
• All positive real numbers have two square roots: a
positive and negative square root.
• 3^2=9 then 3 is a square root of 9.
• A radicand is the number or expression inside a radical
symbol.
• Perfect squares are numbers whose square roots are
integers or quotients of integers.
• An irrational number is a number that cannot be written
as the quotient of two integers.
• - 121=-11 121 is a perfect square: 11^2=121
9.1 Continued
• A quadratic equation is an equation that can be written in
the following standard form.
• ax^2+bx+c=0, where a
0
• In standard form, a is the leading coefficient.
• Solving x^2=d by finding square roots:
• If d>0, then x=d has two solutions: x=+/• If d=0, then x^2=d has one solution: x=0.
• If d<0, then x^2=d has no real solution.
• x^2=4 has two solutions: x= -2, +2
• x^2=0 has one solution: x= 0
• x^2= -1 has no real solution
Falling Object Model: h= -16t^2+s
• h=height, t=time, s=initial height
• s=32; h= -16t^2+32
d.
9.2-Simplifying Radicals
• Product Property: the square root of a product equals the
product of the square roots of the factors.
•
ab= a x
b when a and b are positive numbers.
•
4 x 100= 4 x 100
• Quotient Property: the square root of a quotient equals the
quotient of the square roots of the numerator and denominator.
•
a/b = a/ b when a and b are positive numbers.
•
9/25 = 9/ 25
• An expression with radicals is in simplest form if the following
are true:
• No perfect square factors other than 1 are in the radicand.
• No fractions are in the radicand.
• No radicals appear in the denominator of a fraction.
9.3-Graphing Quadratic Functions
• Every quadratic function has a U-shaped graph called a parabola.
• If the leading coefficient a is positive the parabola opens up, if it’s
negative the parabola opens down.
• The vertex is the lowest point of a parabola that opens up and the
highest point of a parabola that opens down.
• The line passing through the vertex that divides the parabola into two
symmetric parts is called the axis of symmetry.
• Graph of a Quadratic Function: the graph of y=ax^2+bx+c is a parabola.
• The vertex has an x-coordinate –b/2a.
• The axis of symmetry is the vertical line x= -b/2a.
• To Graph:
• Find the x-coordinate of the vertex.
• Make a table of values, using x-values to the left and
right of the vertex.
• Plot the points and connect them with a smooth curve
to form a parabola.
9.4-Solving Quadratic Equations by
Graphing
•
•
•
•
Step 1: Write the equation in the form ax^2+bx+c=0
Step 2: Write the related function y=ax^2+bx+c
Step 3: Sketch the graph of the function y=ax^2+bx+c
The solutions of ax^2+bx+c=0 are the x-intercepts.
9.5-Solving Quadratic Equations by
the Quadratic Formula
• Quadratic Formula:
• Use this formula to solve quadratic
equation: ax^2+bx+c=0
•
9.5 Continued
• Vertical Motion Models:
Object is dropped: h= -16t^2+s
Object is thrown: h= -16t^2+vt+s
• h=height (ft), t=time in motion (sec)
s=initial height (ft), v=initial velocity (ft/sec)
• Example: s=200ft, v= -30ft/sec (the object
thrown)
• h= -16t^2+( -30)t+200
• Substitute 0 for h, write in standard form.
• t= -(-30)+/(-30)^2-4(-16)(200) over 2(-16)
• Simplify: t=30+/- 13,700 over -32
• Solutions: t=2.72 or -4.60
9.6-Applications of the Discriminant
•
:b^2-4ac=discriminant
• The discriminant is used to find the number of solutions of a
quadratic equation.
• Number of solutions of a quadratic equation:
• If discriminant is positive, then 2 solutions.
• If discriminant is 0, then 1 solution.
• If discriminant is negative, then no real solutions.
• Example: x^2-3x-4=0
• b^2-4ac=(-3)^2-4(1)(-4)
=9+16
=25, discriminant is positive. 2 solutions.
9.7- Graphing Quadratic
Inequalities
• If quadratic inequality has < or > then it will have a dotted
parabola.
• If inequality has
or
than it will have a solid lined
parabola.
• Check a point somewhere in the parabola; if it’s a
solution, shade its region. If not, shade the other region.
9.8-Comparing Linear, Exponential,
and Quadratic Models.
• Three basic models:
• Linear= y=mx+b
• Exponential= y=c(1+/-r)^t
• Quadratic= y=ax^2+bx+c
9.8 Continued
(0, 3) (8, 3) (-4, -1) (4, 4) (-6, -3) (10, 1)
• Plot points on graph
• See what shape it gives you, decide
between linear, exponential, and quadratic
models.