Warm Up Grab your calculator Find the vertex: y = 2x x + 53 (-5, 3)
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1 Warm Up Grab your calculator Find the vertex: y = 2x x + 53 (-5, 3)
2 Quiz will be next Tuesday, folks.
3 Check HW/ New Section
4 Another useful form of writing quadratic functions is the standard form. The standard form of a quadratic function is f(x)= ax 2 + bx + c, where a 0. The coefficients a, b, and c can show properties of the graph of the function. You can determine these properties by expanding the vertex form. f(x)= a(x h) 2 + k f(x)= a(x 2 2xh +h 2 ) + k f(x)= a(x 2 ) a(2hx) + a(h 2 ) + k f(x)= ax 2 + ( 2ah)x + (ah 2 + k) Multiply to expand (x h) 2. Distribute a. Simplify and group terms.
5 a = a a in standard form is the same as in vertex form. It indicates whether a reflection and/or vertical stretch or compression has been applied.
6 b = 2ah Solving for h gives. Therefore, the axis of symmetry, x = h, for a quadratic function in standard form is.
7 c = ah 2 + k Notice that the value of c is the same value given by the vertex form of f when x = 0: f(0) = a(0 h) 2 + k = ah 2 + k. So c is the y-intercept.
8
9 Consider the function f(x) = 2x 2 4x + 5. a. Determine whether the graph opens upward or downward. Because a is positive, the parabola opens upward. b. Find the axis of symmetry. The axis of symmetry is given by. Substitute 4 for b and 2 for a. The axis of symmetry is the line x = 1.
10 Consider the function f(x) = 2x 2 4x + 5. 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 5.
11 Consider the function f(x) = x 2 2x + 3. a. Determine whether the graph opens upward or downward. Because a is negative, the parabola opens downward. b. Find the axis of symmetry. The axis of symmetry is given by. Substitute 2 for b and 1 for a. The axis of symmetry is the line x = 1.
12 Consider the function f(x) = x 2 2x + 3. 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 3.
13 For the function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept f(x)= 2x 2 4x a. Because a is negative, the parabola opens downward. b. The axis of symmetry is given by. Substitute 4 for b and 2 for a. The axis of symmetry is the line x = 1.
14 f(x)= 2x 2 4x c. 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) = 2 The vertex is ( 1, 2). d. Because c is 0, the y-intercept is 0.
15 For the function, g(x)= x 2 + 3x 1. (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept a. Because a is positive, the parabola opens upward. b. The axis of symmetry is given by. Substitute 3 for b and 1 for a. The axis of symmetry is the line.
16 g(x)= x 2 + 3x 1 c. The vertex lies on the axis of symmetry, so the x-coordinate is. The y-coordinate is the value of the function at this x-value, or f( ). f( ) = ( ) 2 + 3( ) 1 = The vertex is (, ). d. Because c = 1, the intercept is 1.
17 Find the minimum or maximum value of f(x) = 3x 2 + 2x 4. Then state the domain and range of the function. Step 1 Determine whether the function has minimum or maximum value. Because a is negative, the graph opens downward and has a maximum value. Step 2 Find the x-value of the vertex. Substitute 2 for b and 3 for a.
18 Step 3 Then find the y-value of the vertex, The maximum value is. The domain is all real numbers, R. The range is all real numbers less than or equal to
19 Find the minimum or maximum value of f(x) = x 2 6x + 3. Then state the domain and range of the function. Step 1 Determine whether the function has minimum or maximum value. Because a is positive, the graph opens upward and has a minimum value. Step 2 Find the x-value of the vertex.
20 Step 3 Then find the y-value of the vertex, 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}.
21 Now can we? Find the vertex: y = 2x x + 53 (-5, 3)
22 The average height h in centimeters of a certain type of grain can be modeled by the function h(r) = 0.024r r , where r is the distance in centimeters between the rows in which the grain is planted. Based on this model, what is the minimum average height of the grain, and what is the row spacing that results in this height?
23 The minimum value will be at the vertex (r, h(r)). Step 1 Find the r-value of the vertex using a = and b = 1.28.
24 Step 2 Substitute this r-value into h to find the corresponding minimum, h(r). h(r) = 0.024r r Substitute for r. h(26.67) = 0.024(26.67) (26.67) h(26.67) 16.5 Use a calculator. The minimum height of the grain is about 16.5 cm planted at 26.7 cm apart.
25 Check Graph the function on a graphing calculator. Use the MINIMUM feature under the CALCULATE menu to approximate the minimum. The graph supports the answer.
26 The highway mileage m in miles per gallon for a compact car is approximately by m(s) = 0.025s s 30, where s is the speed in miles per hour. What is the maximum mileage for this compact car to the nearest tenth of a mile per gallon? What speed results in this mileage?
27 The maximum value will be at the vertex (s, m(s)). Step 1 Find the s-value of the vertex using a = and b = s ( 2.45) ( 0.02 ) b = - = - = 2a
28 Step 2 Substitute this s-value into m to find the corresponding maximum, m(s). m(s) = 0.025s s 30 Substitute 49 for r. m(49) = 0.025(49) (49) 30 m(49) 30 Use a calculator. The maximum mileage is 30 mi/gal at 49 mi/h.
29 Check Graph the function on a graphing calculator. Use the MAXIMUM feature under the CALCULATE menu to approximate the MAXIMUM. The graph supports the answer.
30 Homework Worksheet
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