Quadratic Functions

Definition of a Quadratic Function  A quadratic function is defined as: f(x) = ax² + bx + c where a, b and c are real numbers and a ≠0. f(x) = ax² + bx + c where a, b and c are real numbers and a ≠0.  Graphs of quadratic functions are parabolas (u-shaped).

Vertex of a parabola  The vertex (x, y) of a parabola is the turning point of a parabola.  The vertex (x, y) is at a minimum for a regular U-shaped parabola. The y-value of the vertex is the absolute minimum value for the function.  The vertex (x, y) is at a maximum for an “upside down” U shaped parabola. The y-value of the vertex is the absolute maximum value for the function. Vertex/Minimum Vertex/Maximum

Axis of Symmetry  The axis of symmetry (A.O.S.) divides the parabola into 2 mirror image halves. The equation for the AOS is the equation for a vertical line, written as “x =___”.  The x-coordinate of the vertex is the value that you use for the AOS. Axis of Symmetry – Vertical Line ( x = # )

Reflection Points  Reflection points are found at equal distances from the axis of symmetry, or vertex, and always share the same y-values.  To graph a parabola by hand, find the vertex and then choose x values at equal distances on either side of the vertex to evaluate. Vertex/Minimum Reflection Points

Graphing parabolas by hand  To find the x-coordinate of the vertex: Use the formula  Then, substitute the x-coordinate into the equation and evaluate f(x) or y.

Graph: f(x) = 2x²-8x+1  1) Find the x-coordinate of the vertex. Note: a = 2, b = -8 and c = 1  2) Evaluate the function at x = 2 (find f(2))  The coordinates of the vertex are (2, -7)

 Create a table of values with the vertex in the middle of the table. Then, choose x-values on either side of the vertex to graph the parabola.  Evaluate f(0), f(1), f(3) and f(4).  Then graph the parabola and draw the axis of symmetry.  What is the equation for the axis of symmetry?  Is the vertex at a maximum or minimum?  What is the max/min y-value?  What is the domain/range? xy

Task 1: Given two reflection points, find the equation for the line of symmetry  Example: Find the equation for the line of symmetry given two reflection points: (3, -2) and (-8,-2)

Example:  Find the equation for the line of symmetry given two reflection points: (3, -2) and (-8,-2) (3, -2) and (-8,-2) Find the mid point of the segment joining the two reflection points = -5 A.O.S.: x = Reflection Points will have same Y-values. (3, -2) (-8, -2)

Given two reflection points, find equation for line of symmetry  1) (3, 7); (13, 7)  2) (5, 9); (12,9)  3) (-2, -3); (8, -3)  4) (6.4, 5.2); (8.6, 5.2)

Given two reflection points, find line of symmetry-Answers  1) (3, 7); (13, 7) x = 8  2) (5, 9); (12,9) x= 8.5  3) (-2, -3); (8, -3) x=3  4) (6.4, 5.2); (8.6, 5.2) x=7.5

Task 2: Given the vertex (V) and a point on parabola, find another point on the parabola. (Find a reflection point) Example: V:( -2,4) P(1.5,-8) Find another point on the parabola.

 Example: V:( -2,4) P(1.5,-8) Reflection Points Will be at same height Same y value (-2, 4) (1.5, - 8) ( ?, - 8) 3.5 ? = The Reflection Point will be at ( - 5.5, - 8 )

Given vertex and one point, find another point on same parabola.  5) V: (1, 1); P (3, 4)  6) V: (-5, 6); P (0, 5)  7) V: (4, -6); P (-3.2, 11)  8) V: (-3, -4); P (5, 6)

Given vertex and one point, find another point on parabola-Answers  5) V: (1, 1); P (3, 4) P’ = (-1, 4)  6) V: (-5, 6); P (0, 5) P’ = (-10, 5)  7) V: (4, -6); P (-3.2, 11) P’ = (11.2, 11)  8) V: (-3, -4); P (5, 6) P’=( -11, 6)

Algebraically: V:( -2,4) P(1.5,-8) The distance between the given x- coordinates is: The distance between the given x- coordinates is: The point we are looking for is to the left of the vertex, so subtract from -2. The point we are looking for is to the left of the vertex, so subtract from -2. ( = -5.5) The reflection point is (-5.5, -8) The reflection point is (-5.5, -8)

Task: Given the vertex (V) and a point on parabola, find another point  Basically, find the reflection point. To do this: 1) Draw a quick sketch. The y-coordinate of the point you are looking for is the same as the given point. 1) Draw a quick sketch. The y-coordinate of the point you are looking for is the same as the given point. To find the x-coordinate of the reflection point: 2) Determine the distance between the x- coordinates of V and P 2) Determine the distance between the x- coordinates of V and P 3) If reflection point is to left of vertex, subtract from the x-coordinate of V. If reflection point is to right of vertex, add from the x-coordinate of V. 3) If reflection point is to left of vertex, subtract from the x-coordinate of V. If reflection point is to right of vertex, add from the x-coordinate of V.

Task 3: Write a quadratic equation for a parabola with vertex at the origin passing through a given point. Example: (-2, 8)

 Example: ( -2,8) (0, 0) (-2, 8) What is the equation for this parabola if it were reflected over the x-axis?

Task: Write a quadratic equation for a parabola with vertex at the origin passing through a given point, then find equation for the reflection of the parabola.  Try! 1) P:(1,1) 2) P:(1,-4) 3) P: (2, -4) 4) P: (-3, -45)

Quadratic Function-a function in the form where a, b, and c are real numbers and Parabola-the U-shaped (or upside down U) curve that EVERY quadratic makes when graphed Vertex-The lowest or highest point on a parabola (always given as an ordered pair) Minimum/Maximum-The lowest or highest y-value (always the y-value of the vertex) given in the form y = Axis of Symmetry-the vertical line through the vertex that cuts the parabola into two mirror images (always the x-value of the vertex) given in the form x = Image/Reflection Points-points on the parabola that are equidistant from the line of symmetry (given as a coordinate and always have the same y-value) Quadratic Terms

Graphing Quadratic Functions on the Calculator  Graph in y1 = screen and find a good window (you must be able to see the vertex)  To find the coordinates of the vertex: 2 nd Calc Maximum or Minimum 2 nd Calc Maximum or Minimum Move cursor to left side of vertex, Enter Move cursor to left side of vertex, Enter Move cursor to right side of vertex, Enter Move cursor to right side of vertex, Enter Enter Enter

Graphing Quadratic Functions on the Calculator cont.  To find reflection points: Go to Table of Values and find vertex Go to Table of Values and find vertex Look in table on either side of vertex for sets of reflection points Look in table on either side of vertex for sets of reflection points  To find the x-intercepts: Graph the x-axis in Y2 (y = 0) Graph the x-axis in Y2 (y = 0) Note how many times it crosses x-axis Note how many times it crosses x-axis If it does not touch the x-axis, there are no x-intercepts.If it does not touch the x-axis, there are no x-intercepts. If the vertex (x,y) is on the x-axis, there is one x-intercept.If the vertex (x,y) is on the x-axis, there is one x-intercept. Otherwise, there are two x-intercepts.Otherwise, there are two x-intercepts. 2 nd Calc Intersect to find the places where Y1 and Y2 intersect 2 nd Calc Intersect to find the places where Y1 and Y2 intersect Do not move cursor, just push Enter 3 times.Do not move cursor, just push Enter 3 times.

Problem Solving with Quadratic Functions  Quadratic Functions model the path of a falling object.  After t seconds, the height of an object with an initial upward velocity of v 0 meters per second and an initial height of h 0 meters is:  If h 0 is measure in feet and v 0 in feet per second, then the height is:  In each equation, the force of gravity is represented by a squared term in the negative direction. As the time increases, the t 2 term overpowers the t term, and the object falls.

Example: A flea jumps straight up from the ground with an initial upward velocity of 6 feet per second. What will the height of the flea be after 0.2 seconds? Because this problem uses feet as it units, we must use the equation that is written for this…

Another Example: A ball is thrown directly upward from an initial height of 200 feet with and initial velocity of 96 feet per second. After how many seconds will the ball reach its maximum height? And, what is the maximum height? Because this problem uses feet as it units, we must use the equation that is written for this… After 3 seconds, the ball reaches its maximum height.

Calculator Example: Rob’s Football Rob is playing football and he throws the ball from an initial height of 5.5 feet at an initial velocity of 80 feet per second. Write an equation that represents the height of the rocket at any time. Now write the equation as you would enter it on the calculator. What do x and y represent? What will the height of the football be after seconds? What will the height of the football be after 4.75 seconds? After how many seconds will the ball reach its maximum height? What will this maximum height be? How long does it take the ball to hit the ground? How long does it take the ball to initially rise to 60 feet? How long does it take the ball to get to 60 feet on the way down?

Write an equation that represents the height of the rocket at any time. Now write the equation as you would enter it on the calculator. What do x and y represent? Rob is playing football and he throws the ball from an initial height of 5.5 feet at an initial velocity of 80 feet per second. h(t) = -16t t y = -16x x x = time (seconds) y = height (feet)

Graph the function. Note your window size. Rob is playing football and he throws the ball from an initial height of 5.5 feet at an initial velocity of 80 feet per second.

  What will the height of the football be after seconds? The height of the football will be feet.   What will the height of the football be after 4.75 seconds? The height of the football will be 24.5 feet. Rob is playing football and he throws the ball from an initial height of 5.5 feet at an initial velocity of 80 feet per second.

  After how many seconds will the ball reach its maximum height?   What will this maximum height be? Rob is playing football and he throws the ball from an initial height of 5.5 feet at an initial velocity of 80 feet per second. After 2.5 seconds… feet

How long does it take the ball to hit the ground? Rob is playing football and he throws the ball from an initial height of 5.5 feet at an initial velocity of 80 feet per second. It takes 5.07 seconds to hit the ground

  How long does it take the ball to initially rise to 60 feet?   How long does it take the ball to get to 60 feet on the way down? Rob is playing football and he throws the ball from an initial height of 5.5 feet at an initial velocity of 80 feet per second. It takes.81 seconds to initially rise to 60 feet. It takes 4.19 seconds to get to 60 feet on the way down.

Another Calculator Example: Because this problem uses meters as it units, we must use the equation that is written for this… A ball is thrown directly upward from an initial height of 50 meters with an initial velocity of 30 meters per second. After how many seconds will the ball reach its maximum height? What will this maximum height be? After how many seconds will the ball hit the ground? How high is the ball after 2 seconds?