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Properties of Quadratics

Properties of Quadratics. Sec. 12.3. Objectives. I can identify direction of a parabola via equation or graph. I can find the absolute minimum or maximum of a quadratic function. I can find the vertex and axis of symmetry of a quadratic function.

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Properties of Quadratics

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  1. Properties of Quadratics Sec. 12.3

  2. Objectives • I can identify direction of a parabola via equation or graph. • I can find the absolute minimum or maximum of a quadratic function. • I can find the vertex and axis of symmetry of a quadratic function. • I can identify the domain, range, increasing intervals, and decreasing intervals of a quadratic function. • I can find y-int and zeros of a quadratic function via graph.

  3. Quadratic Function Investigation • Now we need to look a little closer at the quadratic graphs themselves. • With a new partner (someone you did not work with already), compare the graphs you have for the quadratic functions in Problem 3. • Make a list of characteristics you notice about these graphs. • Be ready to discuss in 3 minutes!

  4. Today, we are going to investigate quadratic functions.

  5. The Basics • Quadratic Function – a function which has the highest exponent of 2. • Standard Form of a Quadratic Function: , where . • Parabola – the graph created by a quadratic function.

  6. Direction of Parabolas What can you generalize about direction considering the graphs and equations we have already looked at?

  7. Direction of a Parabola

  8. Absolute maximum/minimum • Direction also gives us an important piece of information about the graph. • What else do you notice about the shape of the graphs and direction?

  9. Absolute Minimum/Maximum Absolute Maximum – the highest point on the parabola. Absolute Minimum – the lowest point on the parabola.

  10. Vertex and Axis of Symmetry

  11. Vertex and Axis of Symmetry • The absolute minimum/maximum of a quadratic has another important role in the parabola – the vertex. • Vertex – the central point of a parabola. • Along with the vertex is the axis of symmetry. • Axis of Symmetry (AOS) – the vertical line that passes through the vertex that divides the parabola into 2 mirror images.

  12. Vertex and AOS • Given the graph: • What is the vertex? • What is the AOS? • How could you use the formula to find these features?

  13. Vertex and AOS • Axis of Symmetry: • Vertex - , where h is the AOS of the parabola. • To find the vertex, find the AOS then plug it into the function to find k. , where a and b come from the equation

  14. Vertex and AOS Axis of Symmetry Vertex Does this quadratic have a minimum or a maximum? What is that value?

  15. Try it out! Round 1 • For each function, find the vertex and axis of symmetry. Also, identify the absolute minimum or absolute maximum of each function. 2. 1. AOS: x = -3 Vertex: (-3, 5.5) Maximum: 5.5 AOS: x = -0.75 Vertex: (-0.75, -2.125) Minimum: -2.125

  16. Try it out, part 2! • Identify the direction, AOS, vertex, and min/max value for each function. Direction: UP Direction: Down Direction: UP AOS: x = 2 AOS: x = 0 AOS: x = -1 Vertex: Vertex: Vertex: Minimum: -2 Minimum: 0 Maximum: 3

  17. Intercepts

  18. Intercepts • As with linear functions, x & y intercepts play a big role in the graph. • Your answer should be written as a point. • Y-Intercept – where the graph crosses the y-axis. • What the y-value is when x = 0. • Given a quadratic function, the y-intercept is the c in • X-Intercept – where the graph crosses the x-axis, • Also known as Zeros of a function. • What the x-value is when y = 0. • We will discuss methods of finding zeros in the next lesson.

  19. X & Y Intercepts • Identify the y-intercept of the function. • Identify the x-intercept(s) of the function. • What do you notice? (0, -3) (-1,0) and (3, 0 )

  20. Try it out! Round 1 • Identify the y-intercepts and zero from the graph. Y-intercept: (0, 2) X-intercepts: (1, 0) and (2, 0)

  21. Try it out! Round 2. • Identify the y-intercepts for each function. Y-intercept: (0, -4) Y-intercept: (0,12) How could you find the zeros?

  22. Domain and Range

  23. Domain and Range • Just like with linear functions, we need to look at the domain and range of a quadratic function. • What does it mean to look at the DOMAIN of a function? • What does it mean to look at the RANGE of a function? Domain = x-values of a function Range = y-values of a function

  24. Domain and Range • Consider the linear function • What is it’s domain and range? Domain = All Real Numbers Range = All Real Numbers • Now look at • What‘s different? Domain = All Real Numbers Range =

  25. Domain and Range • Now consider this Quadratic from earlier. • What is the domain? • What is the range? All Real Numbers

  26. Quadratic Domain and Range • Domain – All Real Numbers • Range – depends on the direction of the graph as well as absolute max or min. • Absolute Max = • Absolute Min = • Where k is the y-value of the min/max.

  27. Examples • For each function, identify the domain and range. 2. 1. Domain: Domain: All Real Numbers All Real Numbers -2.125 Range: Range: What do you need to know to identify the range? The vertex

  28. Try it out! 2. • Identify the domain and range for each function. Domain: All Real Numbers Domain: All Real Numbers Range: Range: Domain: All Real Numbers Range:

  29. Interval Notation

  30. Interval Notation • Another way of noting domain and range of a function is to use interval notation. • Intervals – set of real numbers between 2 given numbers. • When talking intervals, we use 2 different notations: • Parenthesis (): excludes the number enclosed in the set. • Bracket []: includes the number enclosed in the set. • If we do not have a bound (start or stop point), we use or . • Only use () with or .

  31. Take a look back at • Domain: • Range: All Real Numbers Interval Notation: Interval Notation:

  32. Take a look back at • We can use interval notation to analyze graphs’ domain, range, intervals of increasing, and intervals of decreasing. • Increasing: where the graph is going up. We refer to the interval using the x values. • Decreasing: where the graph is going down. We refer to the interval using the x values.

  33. Take a look back at • Increasing: • Decreasing: 0]

  34. Try it out: Identify the increasing and decreasing intervals for the function using interval notation. Increasing: 2. Decreasing: Increasing: Decreasing:

  35. Try it out! • For each function, write the domain and range using interval notation. Then, identify the increasing and decreasing intervals using interval notation. 2. 1. Domain: Domain: Range: [ Range: Increasing: [ Increasing: Decreasing: (- Decreasing:

  36. Putting it all together Quadratic Functions

  37. Try b, c, and d with a partner. pg. M3-179 #8 All Real Numbers

  38. All Real Numbers

  39. All Real Numbers

  40. All Real Numbers

  41. Closing • Given the function , find the following characteristics of the graph: 1. Direction 2. Absolute max/min 3. AOS 4. Vertex 5. y-intercept 5. Domain 6. Range 7. Interval of increase and decrease

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