Characteristics of Quadratic Functions Topic 1 Characteristics of Quadratic Functions Unit 6 Topic 1

Explore: Part 1 A quadratic function is an equation written in the standard form 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐, where a≠0. Sketch the following functions with the help of a graphing calculator. Enter the functions into Y= . For each activity below, enter graph 1 into Y1 and graph 2 into Y2. What do you notice? Activity 1 1) 𝑦= 𝑥 2 +3𝑥+2 2) 𝑦= −𝑥 2 +3𝑥+2

Explore: Part 1 Activity 2 1) 𝑦= 2𝑥 2 +5𝑥+1 2) 𝑦= −2𝑥 2 +5𝑥+1 Sketch the following functions with the help of a graphing calculator. Enter the functions into Y= . For each activity below, enter graph 1 into Y1 and graph 2 into Y2. What do you notice? Activity 2 1) 𝑦= 2𝑥 2 +5𝑥+1 2) 𝑦= −2𝑥 2 +5𝑥+1 What happens when the value of a is changed to a negative?

You should notice… When the value of a is changed to a negative, the graph changes – it opens downward. When the a-value is positive, the graph opens up When the a-value is negative, the graph opens down

Explore: Part 2 1) 𝑦= 𝑥 2 +3𝑥+2 2) 𝑦= 𝑥 2 +3𝑥+5 Sketch the following functions with the help of a graphing calculator. Enter the functions into Y= . Enter the graphs into Y1 through Y4. What do you notice? 1) 𝑦= 𝑥 2 +3𝑥+2 2) 𝑦= 𝑥 2 +3𝑥+5        3) 𝑦= 𝑥 2 +3𝑥−2 4) 𝑦= 𝑥 2 +3𝑥−5 What happens when the value of c is changed?

You should notice… The c-value affects where the graph is located The c-value gives the y-intercept

Information A quadratic function ( , where a≠0), is a function with a degree of 2. The graph of a quadratic function is a parabola. Below are examples of what parabolas look like: If a is positive, the graph opens up. If a is negative, the graph opens down. The constant term, c, is the value of the parabola’s y-intercept.

Information Parts of the parabola:

Information vertex: the point at which the quadratic function reaches its maximum or minimum value minimum value: the lowest y-value (for a quadratic function that opens up) maximum value: the highest y­-value (for a quadratic function that opens down) axis of symmetry: the line through the vertex that divides the graph into two symmetric halves; the x-coordinate of the vertex defines the equation of the axis of symmetry (if r and s are the x-intercepts, axis of symmetry = 𝑟+𝑠 2 ). x-intercept: where the graph crosses the x-axis (x, 0) y-intercept: where the graph crosses the y-axis (0, y)

Information   When analyzing a parabola, we also need to look at its domain and range. domain: the set of all possible x­-values, expressed in set notation, {x|x∈R}. range: the set of all possible y-values. If there is a minimum, the range will be {y|y ≥ minimum, y ∈ R} If there is a maximum, the range will be {y|y ≤ maximum, y ∈ R}

Example 1 For each of quadratic function, fill in the characteristics: Try this on your own first!!!! Try this on your own first!!!! Characteristics of Quadratic Functions For each of quadratic function, fill in the characteristics: a) up (0, 0) Minimum at y = 0 x = 0 x = 0 y = 0

Example 1 For each of quadratic function, fill in the characteristics: Characteristics of Quadratic Functions For each of quadratic function, fill in the characteristics: b) down (1, 4) Maximum at y = 4 x = 1 x = -1, 3 y = 3

Example 1 For each of quadratic function, fill in the characteristics: Characteristics of Quadratic Functions For each of quadratic function, fill in the characteristics: c) up (1, -1) Minimum at y = -1 x = 1 x = 0, 2 y = 0

Example 1 For each of quadratic function, fill in the characteristics: Characteristics of Quadratic Functions For each of quadratic function, fill in the characteristics: d) up (3, 7) Minimum at y = 7 x = 3 none y = 25

Example 2 a) Sketch the graph of , using a table of values. Try this on your own first!!!! Sketching a quadratic function using a table of values a) Sketch the graph of , using a table of values. Substitute each value of x into the equation in order to find each corresponding value of y. Each time you get a coordinate pair, plot it on the graph. x y (x, y) -3   -2 -1 1 2 3

Example 2 Try this on your own first!!!! Sketching a quadratic function using a table of values b) Determine the x-intercepts, y-intercept, the equation of the axis of symmetry, the coordinates of the vertex, the domain, and the range. x–intercepts: x = -3, -1 y–intercept: y = 3 axis of symmetry: x = -2 coordinates of the vertex: (-2, -1) domain: range:

Example 3 The quadratic equation has x-intercepts at x=-2 and x=4 Try this on your own first!!!! Determining characteristics of a quadratic function given the x-intercepts The quadratic equation has x-intercepts at x=-2 and x=4 Is the parabola opening up or down? Will it have a maximum or minimum? What is the equation of the axis of symmetry? Since the a-value is positive, we know it is opening up. Since the axis of symmetry goes through the centre of the graph, it is exactly halfway between the two x-intercepts: at x=1.

Example 3 The quadratic equation has x-intercepts at x=-2 and x=4 Try this on your own first!!!! Determining characteristics of a quadratic function given the x-intercepts The quadratic equation has x-intercepts at x=-2 and x=4 What is the coordinate of the vertex? State the domain and range of the function. Since the axis of symmetry has an x-value of 1, the y-value can be determined by substituting 1 in for x.

Information Another way to determine the vertex of a parabola is to use the graphing calculator.

Example 4 Try this on your own first!!!! Determining characteristics of a quadratic function graphically Given the quadratic equation use a graphing calculator to determine the window settings used vertex axis of symmetry domain and range As we proceed through this question, refer to the steps in your workbook.

Example 4: Solution the window settings used  Using the calculator steps, start by entering the equation into the calculator in Y=. Set up the window. It’s usually a good idea to start with x: [-10, 10, 1] and y: [-10, 10, 1] and then see what the graph looks like.  Press graph. the window settings used since the window shows the graph well, we can state the window settings as is. x:[-10, 10, 1] and y: [-10, 10, 1]

Example 4: Solution b) vertex Vertex is a minimum at (2.5, -2.25)  As you can see in the graph, we have a minimum for a vertex. Press 2nd Trace and select 3: minimum. Follow the prompts on your calculator. Move the cursor to the left of the vertex and press Enter. Then move the cursor to the right of the vertex and press Enter. Then press Enter again when you are asked for a guess.  There is a minimum at (2.5, -2.25) Vertex is a minimum at (2.5, -2.25)

Example 4: Solution c) axis of symmetry d) domain and range The axis of symmetry passes through the vertex, so the axis of symmetry is: x = 2.5 d) domain and range The domain is {x|xЄR} and the range is {y|y ≥ -2.25, y ЄR}

Example 5 Try this on your own first!!!! Quadratic function of a ski jump A skier’s jump was recorded in a frame-by-frame analysis. The picture was used to determine the quadratic function that relates the skier’s height above the ground, y, measured in metres, to the time, x, in seconds that the skier was in the air: 𝑦=−4.9 𝑥 2 +15𝑥+1.   Graph the function in your calculator. State the window settings used.

Example 5a: solution Graph the function in your calculator. State the window settings used.  Using the calculator steps, start by entering the equation into the calculator in Y=. Start the window with x: [0, 10, 1] and y: [0, 10, 1]. We don’t include negative values for x or y since x represents time and y represents height, neither of which can be negative.  Obviously this window setting is not ideal. We need to go higher in the y-values and lower in the x-values. Try again.  Press graph.

Example 5a: solution Graph the function in your calculator. State the window settings used. Much better. The windows used are x: [0, 5, 1] and y: [0, 15, 1] Now that we have a good graph to work from the rest becomes pretty straight forward.

Example 5 Try this on your own first!!!! Quadratic function of a ski jump   b) Determine the skier’s maximum height, to the nearest tenth of a metre. c) When, to the nearest tenth of a second, did the skier reach the maximum height?  As you can see in the graph, we have a maximum for a vertex. Press 2nd Trace and select 4: maximum. Follow the prompts on your calculator to find the maximum. Since height is represented by y, the maximum height reached is 12.5m. Time is represented by x. The skier reached his maximum height at at 1.5 seconds.

Example 5 d) Sketch the function. Try this on your own first!!!! Quadratic function of a ski jump  d) Sketch the function. Draw a sketch that looks like your graph on the calculator. Make sure to label the x-axis and the y-axis ! height time

Example 5 Try this on your own first!!!! Quadratic function of a ski jump e) State the range of this function in the context of this situation.      f) What is the y-intercept of the function? What does it represent in this situation? Note: You can calculate this value by pressing 2nd Trace 1: Value and then enter an x-value of 0. You should notice that the answer is the same as the c-value of the equation. Since we cannot have negative height, and we have a maximum height at 12.5m, the range is: {y|0 ≤ y≤12.5, y∈𝑅} The y-intercept occurs where the x-value (time) is 0. This shows the skier’s height at his starting point. The skiers height at the starting point is 1m.

Example 5 Try this on your own first!!!! Quadratic function of a ski jump g) On the next day of training, the skier increased his speed before taking the same jump. The equation that models the skier’s new jump is 𝑦=−4.9 𝑥 2 +17𝑥+1. How much higher did the skier go on this jump, to the nearest tenth of a metre? Start by graphing the new function in Y=. We have to increase our y-maximum a little bit (20) to see the whole graph. Find the maximum. (15.744898 – 12.479592 = 3.265306) The next day, the skier when 3.3 m higher.

Need to Know: A quadratic function is in the form 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐, where a≠0. When a quadratic function is graphed, its graph is a parabola. If a is positive, the graph opens up. If a is negative, the graph opens down. The constant term, c, is the value of the parabola’s y- intercept. Properties of the parabola include its vertex (which is either a minimum or maximum), axis of symmetry, x-intercepts, y-intercept, domain and range. You’re ready! Try the homework from this section.