Objective 1: Identify the characteristics of a quadratic function from its graph

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1 Section 8.2 Quadratic Functions and Their Graphs Definition Quadratic Function A quadratic function is a second-degree polynomial function of the form, where a, b, and c are real numbers and. Every quadratic function has a u-shaped graph called a parabola. Objective 1: Identify the characteristics of a quadratic function from its graph A parabola either opens up or opens down depending on the leading coefficient,. If, as in Figure 1a, the parabola will open up. If, as in Figure 1b, the parabola will open down. If, the graph will be narrower than the graph of. If, the graph will be wider than the graph of Without graphing, determine if the graph of the quadratic function opens up or down. Also determine if the graph will be wider or narrower than the graph of Five basic characteristics of a parabola: 1. Vertex 2. Axis of symmetry 3. y-intercept 4. x-intercept(s) or real zeros 5. Domain and range

2 8.2.5 Use the given graph of a quadratic function to find a. Vertex b. Axis of symmetry c. y-intercept d. x-intercept(s) e. Domain and range Objective 2: Graph quadratic functions by using translations In mathematics, a translation is when every point on a graph is shifted the same distance in the same direction. We now examine translations of parabolas involving vertical or horizontal shifts. Vertical Shifts of Functions If c is a positive real number: The graph of is obtained by shifting the graph of vertically upward k units. The graph of is obtained by shifting the graph of vertically downward k units Horizontal Shifts of Functions If c is a positive real number: The graph of is obtained by shifting the graph of horizontally left h units. The graph of is obtained by shifting the graph of horizontally right h units Sketch the graph of the quadratic function by using translations. Compare the graph to the graph of.

3 Objective 3: Graph quadratic functions of the form 2 Standard Form of a Quadratic Function f ( x) a x h k A quadratic function is in standard form if it is written as 2 The graph is a parabola with vertex ( hk, ). f ( x) a x h k. x h ( hk, ) ( hk, ) x h a 0 a 0 Domain:, Domain:, Range: k, Range:,k The axis of symmetry of the parabola is the vertical line x = h. Given the quadratic function in standard form 2 f ( x) a x h k, answer the following: 1. What are the coordinates of the vertex? 2. For what values of a does the graph open up? Open down? 3. What is the equation of the axis of symmetry? 4. How do you find any x-intercepts? 5. How do you find any y-intercept? 6. State the domain. 7. State the range Given the quadratic function in standard form, find 1-7 above.

4 Objective 4: Find the vertex of a quadratic function by completing the square Writing in Standard Form by Completing the Square Step 1. Group the variable terms together. Step 2. If, factor a out of the variable terms. Step 3. Take half the coefficient of the x-term inside the parentheses, square it, and add it inside the parentheses. Multiply this value by a, then subtract from c. Step 4. The expression inside the parentheses is now a perfect square. Rewrite it as a binomial squared and simplify the constant term outside of the parentheses Write the quadratic function in standard form and find the vertex Write the quadratic function in standard form and find the vertex. Objective 5: Graph quadratic functions of the form by completing the square. Rewrite the function in standard form by completing the square, then answer/do the following 1. What are the coordinates of the vertex? 2. Does the graph open up? Open down? 3. What is the equation of the axis of symmetry? 4. x-intercepts? 5. y-intercept? 6. Graph 7. State the domain. 8. State the range.

5 Rewrite the quadratic function in standard form then answer 1-8 above Rewrite the quadratic function in standard form then answer 1-8 above. Objective 6: Find the vertex of a quadratic function by using the vertex formula Formula for the Vertex of a Parabola Given a quadratic function of the form,, the vertex of the parabola is given by. The axis of symmetry is the vertical line Use the vertex formula to find the vertex of the quadratic function

6 Objective 7: Graph quadratic functions of the form formula by using the vertex Given the quadratic function, answer the following 1. What are the coordinates of the vertex? 2. Does the graph open up? Open down? 3. What is the equation of the axis of symmetry? 4. x-intercepts? 5. y-intercept? 6. Graph 7. State the domain. 8. State the range Use the quadratic function to answer 1-8 above.

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