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Mod 3 Lesson 3

Mod 3 Lesson 3. Analyzing Graphs of Quadratic and Polynomial Functions. Vocabulary. Domain: the set of x-values where the function is defined Range: the set of y-values extracted from the function Vertex: the maximum or minimum of a quadratic function

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Mod 3 Lesson 3

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  1. Mod 3 Lesson 3 Analyzing Graphs of Quadratic and Polynomial Functions

  2. Vocabulary • Domain: the set of x-values where the function is defined • Range: the set of y-values extracted from the function • Vertex: the maximum or minimum of a quadratic function • Local minimum: where the function has the lowest value in a certain region

  3. More Vocab • Local maximum: where the function has the highest value in a certain region • x-intercept: where the graph crosses the x-axis;; y = 0; the solution to the function • y-intercept: where the graph crosses the y-axis; x = 0 • Increasing interval: where the function is increasing from left to right • Decreasing interval: where the function is decreasing from left to right

  4. Interval Notation • 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 • Interval notation

  5. If we wanted to write 4 < x ≤ 30 in interval notation, what would it look like? (4, 30]

  6. Domain and Range • Remember, domain is all of your possible x-values and range is the y-values of the function

  7. Maximum, Minimum, and x-intercepts Refer back to the Mod 3 Lesson 1 notes as to how to find your max and min ordered pairs

  8. Increasing and decreasing • Click on the link below and answer the questions on your notes sheet about increasing and decreasing intervals • Math is fun

  9. Positive and negative • We can tell when the function has positive and negative values by its y-values. • Positive y-values – function is above the x-axis • Negative y-values – function is below the x-axis • You will need to find the x-intercepts of the function to help you identify these intervals

  10. Symmetry • There are many different ways a function can show symmetry. • Quadratic functions have an axis of symmetry – a vertical line that goes through the vertex. • It can be found by using the formula • It is the x-value of the vertex

  11. Finding A.o.S. • Find the axis of symmetry of the function y = x2 – 2x + 5 = - (-2) = 1 2(1) So x = 1 is the axis of symmetry

  12. Other types of symmetry • Even: when the function is symmetric about the y-axis • Algebraically: f(-x) = f(x) • 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 • Odd: when the function is symmetric about the origin • Algebraically: f(-x) = -f(x) • 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

  13. Determine whether the function is even, odd, or neither 1. f(x) = x2 + 2 f(-x) = (-x)2 + 2 = x2 + 2 = f(x) therefore the function is even 2. f(x) = x4 – 2x + 5 f(-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

  14. 3. f(x) = x5 + x3 - 3x f(-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.

  15. Transformations • Graph y = x2 and y = x2 + 2 on the same graph. • What do you notice? • Graph y = x2 and y = (x – 2)2 on the same graph. • What do you notice?

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

  17. Transformations • Graph y = x2 and y = 2x2 on the same graph. • What do you notice? • Graph y = x2 and y = -x2 on the same graph. • What do you notice?

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

  19. Transformations • When we look at these transformations, we can see each piece shifts the graph in a special way. • a: vertically stretches or compresses the graph (a>1 stretch, 0<a<1 compress) • If a is negative, it reflects is over the x-axis • h: shifts the graph left or right (x-h right, x+h left) • k: shifts the graph up or down (+k up, -k down)

  20. Let’s identify the transformations • Quadratic function Cubic function • Vertically compressed reflected over x-axis • Right 1 vertically stretched left 2 down 8

  21. Write the function given the transformations: • Quadratic function shifted 2 units right and 5 units up • Cubic function shifted 3 units left, 7 units down, and reflected about the x-axis

  22. Lets look at an example

  23. Example 2

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