Analyzing Graphs of Quadratic and Polynomial Functions.
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Transcript of Analyzing Graphs of Quadratic and Polynomial Functions.
![Page 1: Analyzing Graphs of Quadratic and Polynomial Functions.](https://reader038.fdocuments.in/reader038/viewer/2022102809/56649f4d5503460f94c6e7e0/html5/thumbnails/1.jpg)
Analyzing Graphs of Quadratic and Polynomial
Functions
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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
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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
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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
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If we wanted to write 4 < x ≤ 30 in interval notation,
what would it look like?
(4, 30]
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Domain and RangeO Remember, domain is all of your
possible x-values and range is the y-values of the function
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Maximum, Minimum, and
x-interceptsRefer back to the Mod 6 Lesson 1 notes as to how to find your max
and min ordered pairs
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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
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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
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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
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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
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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
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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
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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.
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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?
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The graph shifts up 2 units the graph shifts right 2 units
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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?
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The graph is vertically stretched. The graph is reflected over the x-axis.
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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)
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Let’s identify the transformations
O Quadratic function Cubic functionO Vertically compressed reflected over x-
axisO Right 1 vertically
stretchedleft 2down 8
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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
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Lets look at an example
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Example 2