College Algebra. Quadratic Functions and their Graphs. Dr. Nguyen October 12, Department of Mathematics UK

Transcription

1 College Algebra Quadratic Functions and their Graphs Dr. Nguyen Department of Mathematics UK October 12, 2018

2 Agenda Quadratic functions and their graphs Parabolas and vertices Standard form Writing equations from two points Application: nding maximum/minimum values

3 Quadratic Functions A quadratic function is a function that is equivalent to a function of the form q(x) = ax 2 + bx + c, where a, b, and c are constants and a 0. We'll call this the general form of a quadratic function. The graph of a quadratic function is a parabola. The tip of the parabola is the vertex. The most basic quadratic function is p(x) = x 2. Its vertex is (0,0). It opens up.

4 Quadratic Functions A quadratic function is a function that is equivalent to a function of the form q(x) = ax 2 + bx + c, where a, b, and c are constants and a 0. We'll call this the general form of a quadratic function. The graph of a quadratic function is a parabola. The tip of the parabola is the vertex. The most basic quadratic function is p(x) = x 2. Its vertex is (0,0). It opens up.

5 Quadratic Functions A quadratic function is a function that is equivalent to a function of the form q(x) = ax 2 + bx + c, where a, b, and c are constants and a 0. We'll call this the general form of a quadratic function. The graph of a quadratic function is a parabola. The tip of the parabola is the vertex. The most basic quadratic function is p(x) = x 2. Its vertex is (0,0). It opens up.

6 Quadratic Functions A quadratic function is a function that is equivalent to a function of the form q(x) = ax 2 + bx + c, where a, b, and c are constants and a 0. We'll call this the general form of a quadratic function. The graph of a quadratic function is a parabola. The tip of the parabola is the vertex. The most basic quadratic function is p(x) = x 2. Its vertex is (0,0). It opens up.

7 Graph of p(x) = x 2

8 Standard Form of a Quadratic Function A quadratic function q(x) = ax 2 + bx + c can be rewritten in standard form: q(x) = a(x h) 2 + k, by completing the square. The vertex is easier to nd using standard form: the vertex is (h,k).

9 Standard Form of a Quadratic Function A quadratic function q(x) = ax 2 + bx + c can be rewritten in standard form: q(x) = a(x h) 2 + k, by completing the square. The vertex is easier to nd using standard form: the vertex is (h,k).

10 Ex1: Standard Form of a Quadratic Function A quadratic function q(x) = ax 2 + bx + c can be rewritten in standard form: q(x) = a(x h) 2 + k, by completing the square. The vertex is easier to nd using standard form: the vertex is (h,k). Example: the vertex of is (3,4). q(x) = 2(x 3) 2 + 4

11 Parabolas and Transformations Let p(x) = x 2. Then the graph of any other quadratic function is a transformation of the graph of p: q(x) = a(x h) 2 + k = a p(x h) + k. h is a horizontal shift. a changes the shape with a vertical scaling. k is the vertical shift. The two shifts move the vertex to (h,k).

12 Parabolas and Transformations Let p(x) = x 2. Then the graph of any other quadratic function is a transformation of the graph of p: q(x) = a(x h) 2 + k = a p(x h) + k. h is a horizontal shift. a changes the shape with a vertical scaling. k is the vertical shift. The two shifts move the vertex to (h,k).

13 Parabolas and Transformations Let p(x) = x 2. Then the graph of any other quadratic function is a transformation of the graph of p: q(x) = a(x h) 2 + k = a p(x h) + k. h is a horizontal shift. a changes the shape with a vertical scaling. k is the vertical shift. The two shifts move the vertex to (h,k).

14 Parabolas and Transformations Let p(x) = x 2. Then the graph of any other quadratic function is a transformation of the graph of p: q(x) = a(x h) 2 + k = a p(x h) + k. h is a horizontal shift. a changes the shape with a vertical scaling. k is the vertical shift. The two shifts move the vertex to (h,k).

15 Parabolas and Transformations Let p(x) = x 2. Then the graph of any other quadratic function is a transformation of the graph of p: q(x) = a(x h) 2 + k = a p(x h) + k. h is a horizontal shift. a changes the shape with a vertical scaling. k is the vertical shift. The two shifts move the vertex to (h,k).

16 Parabolas and Transformations Let p(x) = x 2. Then the graph of any other quadratic function is a transformation of the graph of p: q(x) = a(x h) 2 + k = a p(x h) + k. h is a horizontal shift. a changes the shape with a vertical scaling. k is the vertical shift. The two shifts move the vertex to (h,k).

17 Graph of p(x) = x 2 and q(x) = 2(x 3) Cyan arrows: shifts move vertex from (0,0) to (3,4).

18 Absolute Max/Min Values Let f be a function. The absolute maximum value of f is the largest output y value. The absolute minimum value of f is the smallest output y value. For a quadratic function, the absolute maximum or minimum value is the y-coordinate of the vertex, and It is said to occur at the x-coordinate of the vertex.

19 Absolute Max/Min Values Let f be a function. The absolute maximum value of f is the largest output y value. The absolute minimum value of f is the smallest output y value. For a quadratic function, the absolute maximum or minimum value is the y-coordinate of the vertex, and It is said to occur at the x-coordinate of the vertex.

20 Absolute Max/Min Values Let f be a function. The absolute maximum value of f is the largest output y value. The absolute minimum value of f is the smallest output y value. For a quadratic function, the absolute maximum or minimum value is the y-coordinate of the vertex It is said to occur at the x-coordinate of the vertex.

21 Absolute Max/Min Values Let f be a function. The absolute maximum value of f is the largest output y value. The absolute minimum value of f is the smallest output y value. For a quadratic function, the absolute maximum or minimum value is the y-coordinate of the vertex It is said to occur at the x-coordinate of the vertex.

22 The Shape of a Parabola Let q(x) = a(x h) 2 + k. A parabola's shape is determined by the sign of a: If a > 0, the parabola opens up, and q has an absolute minimum value. If a < 0, the parabola opens down, and q has an absolute maximum value.

23 The Shape of a Parabola Let q(x) = a(x h) 2 + k. A parabola's shape is determined by the sign of a: If a > 0, the parabola opens up, and q has an absolute minimum value. If a < 0, the parabola opens down, and q has an absolute maximum value.

24 The Shape of a Parabola Let q(x) = a(x h) 2 + k. A parabola's shape is determined by the sign of a: If a > 0, the parabola opens up, and q has an absolute minimum value. If a < 0, the parabola opens down, and q has an absolute maximum value.

25 The Shape of a Parabola Let q(x) = a(x h) 2 + k. A parabola's shape is determined by the sign of a: If a > 0, the parabola opens up, and q has an absolute minimum value. If a < 0, the parabola opens down, and q has an absolute maximum value.

26 Ex2: Find Vertex, Shape, Maximum Find the vertex, shape, and absolute maximum value of q(x) = 7x 2 42x + 5.

27 Ex2: Find Vertex, Shape, Maximum Find the vertex, shape, and absolute maximum value of q(x) = 7x 2 42x + 5. Convert to standard form - complete the square. Factor out 7 from x 2 and x terms: = 7(x 2 + 6x) + 5 Inside parentheses, add and subtract ( ) 2 6 = (3) 2 : 2 = 7(x 2 + 6x + (3) 2 (3) 2 ) + 5 = 7((x + 3) 2 9) + 5

28 Ex2: Find Vertex, Shape, Maximum Distribute the 7: = 7(x + 3) = 7(x ( 3)) The vertex is at ( 3,68). Since the coecient a = 7 is negative, the parabola opens down. The absolute maximum value is the y-coordinate of the vertex, 68.

29 C1: Find Vertex Find the vertex of Type and send a point. q(x) = x 2 10x + 4.

30 C1: Find Vertex Find the vertex of Add and subtract q(x) = x 2 10x + 4. ( ) 2 10 = ( 5) 2 : 2 = x 2 10x + ( 5) 2 ( 5) = (x 5) = (x (5)) The vertex is at (5, 21).

31 Ex3: Find Formula Find the rule of the quadratic function whose graph has its vertex at ( 2,3) and passes through the point (5, 95).

32 Ex3: Find Formula Find the rule of the quadratic function whose graph has its vertex at ( 2,3) and passes through the point (5, 95). Use standard form: f (x) = a(x h) 2 + k. Vertex coordinates give h = 2 and k = 3: f (x) = a(x ( 2)) = a(x + 2) Solve for a by plugging in other point: f (5) = 95, so f (5) = a((5) + 2) = a(7) = 49a 2 = a

33 Ex3: Find Formula We got a = 2, h = 2, k = 3. Hence the rule of f in standard form is f (x) = 2(x + 2) To write in general form, use FOIL: f (x) = 2(x + 2)(x + 2) + 3 = 2(x 2 + 4x + 4) + 3 = 2x 2 8x = 2x 2 8x 5.

34 Vertex Coordinates (Quick) If you have a quadratic function q(x) = ax 2 + bx + c in general form, the coordinates (h,k) of the vertex are: h = b 2a, k = q (h).

35 Ex4: Peak of a Thrown Object If you have a quadratic function q(x) = ax 2 + bx + c in general form, the coordinates (h,k) of the vertex are: h = b 2a, k = q (h). John throws a ball. Its height above the ground (in feet) is given by the equation q(x) = 3x x + 5 where x is the distance (in feet) from John to a point on the ground directly below the ball. How far from John is the ball when it reaches the highest point on its ight? How high is the ball at that point?

36 Ex4: Peak of a Thrown Object John throws a ball. Its height above the ground (in feet) is given by the equation q(x) = 3x x + 5 Task: nd the x and y-coordinates of the vertex. Vertex x-coordinate: x = 24 2( 3) = 24 6 = 4. Vertex y-coordinate: y = q(4) = 3(4) (4) + 5 = 3(16) = = 53. The ball is 4 feet from John, and it is 53 feet high in the air.

37 Next Time Please read Section 5.1 of your textbook. We will go over some problems from the practice exam.