7.1A Investigating Quadratic Functions in Vertex (Standard) Form: y = a(x±p) 2 ±q. Parabolas have a, a middle point. For

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1 7.1A Investigating Quadratic Functions in Vertex (Standard) Form: y = a(x±p) ±q y x Graph y x using a table of values x Graph Shape: the graph shape is called a and occurs when the equation has an. y Quick way to graph: Use a basic count: Start at vertex: in this case (0,0) Over 1, back to vertex Over, back to vertex Over 3, Parabolas have a, a middle point. For y x, it is Parabolas have an AXIS OF SYMMETRY, a reflection line that splits the parabola into. It can be shown with a dashed line. In this example, the equation of the axis of symmetry is Parabolas open or. If they open upwards, they go up forever and ever, but only go down so far. Therefore, they have a value. In the example above, the minimum value is. If they open downwards, they go down forever, but only go up so far. Therefore, they have a value. For any graph, you can find the. How far left does the graph go? How far right? In this example, For any graph, you can find the. How far up does the graph go? How far down? In this example,

2 A quadratic function is a function that has a second degree polynomial (has an x term, but nothing higher. The graph shape that results is a PARABOLA. Examples: *Note: f(x) is the same as y q value y = x ± q a) Graph y = x using the basic count: Start at (0,0) and go over 1, over, over 3, b) Graph y x 4 using a table of values: x y Notice: c) Graph y x 3 by count method: q value is: Vertex is: Then do basic count: Vertex: A of S eqn: Vertex:A of S eqn: y = x ± q The q value:

3 a) Graph y = x using the count p value y = (x±p) b)graph y x 4 table of values x y using a Notice: p value Mental Switch: Vertex: A of S eqn: y = (x±p) ±q: c) Graph y x 3 using the count method: Vertex: A of S eqn: Vertex Notes: Practice Example - Graph y = (x + ) 5 using the count method Vertex: A of S eqn:

4 7.1B Investigating Quadratic Functions in Vertex (Standard) Form: y = a(x±p) ±q a value y = ax a) Graph y = x using the count. b) Graph y x using a table of values x y Notice: 1 c) Graph y x using the count method: The a value: Graph y x using a table of values x y Vertex: A of S eqn:

5 The a value: Graph y x using the count method Standard Form: Notes: Graph a) f ( x) x and b) y 4 x 5 For each, find the - vertex - axis of sym eqn - max/min - domain - range a)vertex: A of S eqn: b) Vertex: A of S eqn:

6 7. General Form of a Quadratic Function: ax ± bx ± c In many instances, quadratic functions are not displayed in Vertex (Standard) Form: y = a(x ± p) ± q. Another way quadratic functions can be displayed is in General Form: ax ± bx ± c. Trying to graph equations in General Form is more difficult, as the vertex isn t visually apparent. However, the conversion of a vertex (standard) form function to a general form function, as shown on p.39, gives a formulaic way to get the vertex when in General Form. When in General Form, the vertex (p, q) can be found by: p = q = The a value is the coefficient of the x term. Example 1 Determine the vertex and a value of f(x) = x 4x 3. Then graph and determine the axis of symmetry equation.

7 Example - Determine the vertex and a value of f(x) = 1 x + x +. Then graph and determine the axis of symmetry equation, max/min, domain, and range. Example 3 Without graphing, given that f(x) is a quadratic function with minimum f(1) = -3, find the vertex, axis of symmetry equation, domain, and range. Example 4 Determine a quadratic function with a vertex of (, 1) and a y-intercept of -3. Then, find another point on the function.

8 7.3 Solving Quadratic Equations by Factoring when a = 1 Quadratic functions are written as f(x) = x x 6 OR y = x x 6. A very important feature of the parabola that results from a quadratic function is where it touches or crosses the x-axis. These are the x-intercepts of the parabola. What is the y value at an x-intercept? Therefore, to find the x-intercepts of a parabola, we can set y = 0 (or f(x) = 0) and solve the resulting equation. When y is set to 0, we call the question a quadratic equation instead of a quadratic function. Quadratic function: f(x) = x x 6 OR y = x x 6 Quadratic equation: 0 = x x 6 OR x x 6 = 0 The x-intercepts of the parabola are the zeros of the quadratic function. They are also called the solutions or roots of the quadratic equation. How many x-intercepts can a parabola have? Draw the possibilities: Therefore, how many zeros can there be for a quadratic function? How many roots or solutions can there be for a quadratic equation? finding roots by graphing One method we can use to find the zeros of a quadratic function is to graph a function. Example 1 What are the roots of the equation x 6x + 5 = 0? 1) Change the quadratic equation to a quadratic function. ) Find the vertex and the a value. Graph the function 3) Note any x-intercepts. These are the zeros of the function and the roots of the equation. 4) Check the solution(s).

9 You can often find the roots of a quadratic equation by factoring when in general form ax bx c 0. Example - Solve and check x 3x To solve a quadratic equation means to find the roots. Steps are as follows: 1) Get everything to one side so that only zero is on the other. ) Identify a, b, and c values. 3) If a = 1, find two numbers that multiply to equal the c value that also add to equal the b value. 4) Factor the trinomial into two binomials using the two numbers from step 3. 5) The roots are the x-values that will make the product of the two binomials zero. If either of the binomials equal zero, then the product of the binomials will equal zero. Therefore, identify the x values that make each binomial equal to zero. Check: Sketch the Graph: Example : a) Solve x 8x 40 8 b) Solve x + 6x 108 = 0 Example 3: a) Solve x 5 0. b) Solve x + 3x 8 = 0 Example 4: a) Solve (x 3)(x + 4) = 8 b) Solve x 6x = 0

10 7.4 The Quadratic Formula Quadratic equations can be solved by graphing (4.1), factoring (4.), other methods not being taught in this course, and the Quadratic Formula. Each method has its advantages and limitations. Any quadratic equation can be solved using something called the quadratic formula. When the quadratic equation is in general form (ax ± bx ± c = 0), the quadratic formula is: x = b ± b 4ac a Example 1 Solve using the quadratic formula. Give roots to the nearest hundredth. a) x = x + 1 b) 3x 5x = 9 c) 1 4 x 3x + 9 = 0 d) x 6x + 9 = 0 e) x(x 4) = 33

11 discriminant There could be 0, 1, or resulting roots, depending on the discriminant, the expression under the square root(b 4ac). How would the discriminant determine the number of roots (0, 1, or )? If b 4ac > 0, If b 4ac = 0, If b 4ac < 0, Example Find out how many roots the quadratic equation will have, then solve for the roots to the nearest hundredth. x 1 = 5x

12 7.5A - Quadratic Word Problems Part 1 Example - The path of a rocket fired over a lake is described by the function h(t) = 4.9(t 5) + 14 where h(t) is the height of the rocket, in metres, and t is time in seconds, since the rocket was fired. a) What is the maximum height reached by the rocket? How many seconds after it was fired did the rocket reach this height? b) How high was the rocket above the lake when it was fired? c) At what time does the rocket hit the ground? d) What domain and range are appropriate in this situation? e) How high was the rocket after 7s? Was it on its way up or down?

13 Example A diver jumps from a 3-m springboard with an initial vertical velocity of 6.8m/s. Her height, h, in metres, above the water t seconds after leaving the diving board can be modeled by the functionh(t) = 4.9t + 6.8t + 3. a) What does the y-intercept represent? b) What is the height of the diver 0.6s after leaving the board? c) What maximum height does the diver reach? When does she reach that height? d) How long does it take before the diver hits the water? e) What domain and range are appropriate in this situation?

14 7.5B Quadratic Word Problems Part Example 1 - The area of a rectangular Ping-Pong table is 45ft. The length is 4ft more than the width. What are the dimensions of the table? Example A picture measures 30cm by 1cm. You crop the picture by removing strips of the same width from the top and one side of the picture. This reduces the area to 40% of the original area. Determine the width of the removed strips.

15 Example 3 - Butchart Gardens wants to build a pathway around its rose garden. The rose garden is currently 30m x 0m. The pathway will be built by extending each side by an equal amount. If the area of the garden and path together is times larger than the area of just the rose garden, how wide will the new path be? Example 4 The length of a lacrosse field is 10m less than twice the width. The area of the field is 6600m. Determine the dimensions of an outdoor lacrosse field.