Analyzing Graphs of Quadratic and Polynomial Functions

Analyzing Graphs of Quadratic and Polynomial

Functions

VocabularyO Domain: the set of x-values where

the function is definedO Range: the set of y-values extracted

from the functionO Vertex: the maximum or minimum

of a quadratic functionO Local minimum: where the function

has the lowest value in a certain region

More VocabO Local maximum: where the function has

the highest value in a certain regionO x-intercept: where the graph crosses the

x-axis;; y = 0; the solution to the functionO y-intercept: where the graph crosses the

y-axis; x = 0O Increasing interval: where the function is

increasing from left to rightO Decreasing interval: where the function

is decreasing from left to right

Interval NotationO Instead of writing the intervals using

inequalities, we can use interval notation. Click on the link to learn more about interval notation and how it compares to inequalities

O Interval notation

If we wanted to write 4 < x ≤ 30 in interval notation,

what would it look like? 

(4, 30]

Domain and RangeO Remember, domain is all of your

possible x-values and range is the y-values of the function

Maximum, Minimum, and

x-interceptsRefer back to the Mod 6 Lesson 1 notes as to how to find your max

and min ordered pairs

Increasing and decreasing

O Click on the link below and answer the questions on your notes sheet about increasing and decreasing intervals

OMath is fun

Positive and negativeO We can tell when the function has

positive and negative values by its y-values.

O Positive y-values – function is above the x-axis

O Negative y-values – function is below the x-axis

O You will need to find the x-intercepts of the function to help you identify these intervals

SymmetryO There are many different ways a

function can show symmetry.

O Quadratic functions have an axis of symmetry – a vertical line that goes through the vertex.O It can be found by using the formulaO It is the x-value of the vertex

Finding A.o.S.O Find the axis of symmetry of the

functiony = x2 – 2x + 5

= - (-2) = 1 2(1)

So x = 1 is the axis of symmetry

Other types of symmetry

O Even: when the function is symmetric about the y-axisO Algebraically: f(-x) = f(x)

O This means when you plug in a negative x-value, you get the same y-value as if you plugged in the positive x-value

O Odd: when the function is symmetric about the originO Algebraically: f(-x) = -f(x)

O This means when you plug in a negative x-value, you get the opposite sign of the y-value as if you plugged in the positive x-value

Determine whether the function is even, odd, or

neither1. f(x) = x2 + 2f(-x) = (-x)2 + 2 = x2 + 2 = f(x) therefore the function is even

2. f(x) = x4 – 2x + 5f(-x) = (-x)4 – 2(-x) + 5 = x4 + 2x + 5 this is not f(x) nor –f(x) so this function is neither even nor odd

3. f(x) = x5 + x3 - 3xf(-x) = (-x)5 + (-x)3 – 3(-x) = -x5 – x3 + 3x = - f(x) so the function is odd

DO NOT assume you can tell even or odd by the degree of the polynomial.

TransformationsO Graph y = x2 and y = x2 + 2 on the same graph.

O What do you notice?

O Graph y = x2 and y = (x – 2)2 on the same graph.

O What do you notice?

The graph shifts up 2 units the graph shifts right 2 units

TransformationsO Graph y = x2 and y = 2x2 on the same graph.

O What do you notice?

O Graph y = x2 and y = -x2 on the same graph.

O What do you notice?

The graph is vertically stretched. The graph is reflected over the x-axis.

TransformationsO When we look at these transformations, we can

see each piece shifts the graph in a special way.

O a: vertically stretches or compresses the graph (a>1 stretch, 0<a<1 compress)

O If a is negative, it reflects is over the x-axisO h: shifts the graph left or right (x-h right, x+h

left)O k: shifts the graph up or down (+k up, -k down)

Let’s identify the transformations

O Quadratic function Cubic functionO Vertically compressed reflected over x-

axisO Right 1 vertically

stretchedleft 2down 8

Write the function given the transformations:

O Quadratic function shifted 2 units right and 5 units up

O Cubic function shifted 3 units left, 7 units down, and reflected about the x-axis

Lets look at an example

Example 2