Mod 3 Lesson 3
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Transcript of Mod 3 Lesson 3
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
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 3 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?
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?
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