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Holt Algebra Properties of Quadratic Functions in Standard Form axis of symmetry standard form minimum value maximum value Vocabulary

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Holt Algebra Properties of Quadratic Functions in Standard Form Notes A. Determine whether the graph opens up or downward. B. Find the axis of symmetry. C. Find the vertex. D. Identify the max or min value of the function. E. Find the y-intercept. F. Graph the function. G. State the domain and range of the function. 2. Consider the function f(x)= 2x 2 + 6x – State the shifts and sketch. Name domain and range. A) f(x)= -x B) f(x)= -2x 2 + 4

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Holt Algebra Properties of Quadratic Functions in Standard Form This shows that parabolas are symmetric curves. The axis of symmetry is the line through the vertex of a parabola that divides the parabola into two congruent halves.

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Holt Algebra Properties of Quadratic Functions in Standard Form Example 1: Identifying the Axis of Symmetry Identify the axis of symmetry for the graph of. Because h = –5, the axis of symmetry is the vertical line x = –5.

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Holt Algebra Properties of Quadratic Functions in Standard Form a in standard form is the same as in vertex form. In standard form, the axis of symmetry is x = -b/a In standard form, the y-intercept is (0,c)

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Holt Algebra Properties of Quadratic Functions in Standard Form These properties can be generalized to help you graph quadratic functions.

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Holt Algebra Properties of Quadratic Functions in Standard Form Consider the function f(x) = 2x 2 – 4x + 5. Example 2A: Graphing Quadratic Functions in Standard Form a. Determine whether the graph opens upward or downward. b. Find the axis of symmetry. Because a is positive, the parabola opens upward. The axis of symmetry is the line x = 1. Substitute –4 for b and 2 for a. The axis of symmetry is given by.

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Holt Algebra Properties of Quadratic Functions in Standard Form Consider the function f(x) = 2x 2 – 4x + 5. Example 2A: Graphing Quadratic Functions in Standard Form c. Find the vertex. The vertex lies on the axis of symmetry, so the x-coordinate is 1. The y-coordinate is the value of the function at this x-value, or f(1). f(1) = 2(1) 2 – 4(1) + 5 = 3 The vertex is (1, 3). d. Find the y-intercept. Because c = 5, the intercept is (0,5).

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Holt Algebra Properties of Quadratic Functions in Standard Form Consider the function f(x) = 2x 2 – 4x + 5. Example 2A: Graphing Quadratic Functions in Standard Form e. Graph the function. Graph by sketching the axis of symmetry and then plotting the vertex and the intercept point (0, 5).

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Holt Algebra Properties of Quadratic Functions in Standard Form Consider the function f(x) = –x 2 – 2x + 3. Example 2B: Graphing Quadratic Functions in Standard Form a. Determine whether the graph opens upward or downward. b. Find the axis of symmetry. Because a is negative, the parabola opens downward. The axis of symmetry is the line x = –1. Substitute –2 for b and –1 for a. The axis of symmetry is given by.

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Holt Algebra Properties of Quadratic Functions in Standard Form Example 2B: Graphing Quadratic Functions in Standard Form c. Find the vertex. The vertex lies on the axis of symmetry, so the x-coordinate is –1. The y-coordinate is the value of the function at this x-value, or f(–1). f(–1) = –(–1) 2 – 2(–1) + 3 = 4 The vertex is (–1, 4). d. Find the y-intercept. Because c = 3, the y-intercept is (0,3). Consider the function f(x) = –x 2 – 2x + 3.

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Holt Algebra Properties of Quadratic Functions in Standard Form Example 2B: Graphing Quadratic Functions in Standard Form e. Graph the function. Graph by sketching the axis of symmetry and then plotting the vertex and the intercept point (0, 3). Consider the function f(x) = –x 2 – 2x + 3.

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Holt Algebra Properties of Quadratic Functions in Standard Form

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Holt Algebra Properties of Quadratic Functions in Standard Form Find the minimum or maximum value of f(x) = –3x 2 + 2x – 4. Then state the domain and range of the function. Example 3: Finding Minimum or Maximum Values Step 1 Determine whether the function has minimum or maximum value. Step 2 Find the x-value of the vertex. Substitute 2 for b and –3 for a. Because a is negative, the graph opens downward and has a maximum value.

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Holt Algebra Properties of Quadratic Functions in Standard Form The maximum value is. The domain is all real numbers, R. The range is all real numbers less than or equal to Example 3 Continued Step 3 Then find the y-value of the vertex, Find the minimum or maximum value of f(x) = –3x 2 + 2x – 4. Then state the domain and range of the function.

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Holt Algebra Properties of Quadratic Functions in Standard Form Find the minimum or maximum value of f(x) = x 2 – 6x + 3. Then state the domain and range of the function. Example 3B Step 1 Determine whether the function has minimum or maximum value. Step 2 Find the x-value of the vertex. Because a is positive, the graph opens upward and has a minimum value.

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Holt Algebra Properties of Quadratic Functions in Standard Form Step 3 Then find the y-value of the vertex, Find the minimum or maximum value of f(x) = x 2 – 6x + 3. Then state the domain and range of the function. f(3) = (3) 2 – 6(3) + 3 = –6 The minimum value is –6. The domain is all real numbers, R. The range is all real numbers greater than or equal to –6, or {y|y –6}. Example 3B Continued

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Holt Algebra Properties of Quadratic Functions in Standard Form Notes A. Determine whether the graph opens upward or downward. B. Find the axis of symmetry. C. Find the vertex. D. Identify the maximum or minimum value of the function. E. Find the y-intercept. x = –1.5 upward (–1.5, –11.5) 2. Consider the function f(x)= 2x 2 + 6x – 7. min.: –11.5 (0,–7)

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Holt Algebra Properties of Quadratic Functions in Standard Form Notes (continued) Consider the function f(x)= 2x 2 + 6x – 7. F. Graph the function. G. Find the domain and range of the function. D: All real numbers ; R {y|y –11.5}

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