Working with Quadratic Functions in Standard and Vertex Forms Example 1: Identify Characteristics of a Quadratic Function in Standard Form f( x) ax bx c, a 0 For the quadratic function f( x) x x 3, identify the direction of opening, the y-intercept and another point that is equidistant from the axis of symmetry, the x-intercepts, the equation of the axis of symmetry, the coordinates of the vertex, the maximum or minimum value, the domain and range. Verify that these characteristics correspond with the graph of the function. direction of opening y-intercept another point which is the same distance from the axis of symmetry as the y-int f( x) x x 3 x-intercepts axis of symmetry vertex maximum or minimum value domain range graph
Example : Analyzing a Quadratic Function in Standard Form A Frisbee is thrown into the air and its height in metres as a function of time in seconds is given by the equation h( t) t 4t 6. Algebraically determine the answers the following questions, then verify your answers on the graph of h(t). a. Determine the initial height of the Frisbee. b. How high is the Frisbee after seconds? c. When does the Frisbee hit the ground? d. When does the Frisbee reach its maximum height? What is this height? e. What are the domain and range in this situation?
Example 3: Convert from Vertex Form to Standard Form Rewrite each function in standard form. a. f ( x ) ( x 3) 5 1 b. f ( x ) ( x 6) 1 Example 4: Convert from Standard Form to Vertex Form when a = 1 Rewrite each function in vertex form and state the vertex for each. a. 1 0 Add and subtract the square of half the coefficient of the x-term to create a perfect square trinomial. Factor the perfect square trinomial (the first three terms). Simplify. This is vertex form. The vertex is at (, ) b. 5 1 Add and subtract the square of half the coefficient of the x-term to create a perfect square trinomial. Factor the perfect square trinomial (the first three terms). Simplify. This is vertex form. The vertex is at (, )
Example 5: Convert from Standard Form to Vertex Form when a 1 Rewrite each function in vertex form and state the vertex for each. a. 1 1 Factor out the leading coefficient from the first two terms. Add and subtract the square of half the coefficient of the x- term to create a perfect square trinomial. Factor the perfect square trinomial (the first three terms). Remember that the distributive property applies to the fourth term in the parentheses. Simplify. This is vertex form. The vertex is at (, ) b. 4 3 1 Factor out the leading coefficient from the first two terms. Add and subtract the square of half the coefficient of the x- term to create a perfect square trinomial. Factor the perfect square trinomial (the first three terms). Remember that the distributive property applies to the fourth term in the parentheses. Simplify. This is vertex form. The vertex is at (, ) c. 3 9 1 Factor out the leading coefficient from the first two terms. Add and subtract the square of half the coefficient of the x- term to create a perfect square trinomial. Factor the perfect square trinomial (the first three terms). Remember that the distributive property applies to the fourth term in the parentheses. Simplify. This is vertex form. The vertex is at (, )
Example 6: Write a Quadratic Function to Model a Situation A student group sells memberships for $6 per year and has 700 members. To increase revenue, the group decides to increase the membership cost. The results of a survey indicate that for every $1 increase in membership cost, 50 fewer students will buy a membership. a. Write a quadratic function in standard form to model this situation. b. Convert the standard form of the function to vertex form. c. What are the coordinates of the vertex? What information does this give the student group? d. Determine the x-intercepts of the function. Which x-intercept is relevant in this situation and what does it represent? e. What are the domain and range in this situation? Solution: a. Let x represent the number of $1 price increases and R(x) represent the revenue. R(x) = (membership cost)(number of members) b. R x x x ( ) 50 400 400 c. The coordinates of the vertex are (, ). This means that a price increase of, to a new price of, will give a maximum revenue of. d. To determine the x-intercepts, let R(x) = 0 and solve for x: The x-intercept of indicates that if the membership cost is increased by, then there would be no revenue (since there would be no memberships sold at this price). e. Domain Range
Example 7: Determine the Relationship Between the Values of p and q in Vertex Form and a, b, and c in Standard Form Expand the vertex form f ( x) a( x p) q and compare the resulting coefficients with the standard form f ( x) ax bx c to see the relationship between the parameters of the two forms of a quadratic function. Solution: f ( x) a( x p) q By comparing the two forms, we can see that b = p = c = q = So the vertex coordinates (p, q) from the vertex form can be determined from standard form b b using p, q, c. a 4a
Example 8: Write a Quadratic Function to Model a Situation A rancher has 100 m of fencing available to build a rectangular corral. a. Write the area of the corral, as a function of the width, in standard form. b. What are the coordinates of the vertex? What do the coordinates of the vertex represent in this situation? c. Determine the domain and range for this situation. Solution: a. Let l represent the length, w represent the width, and A represent the area of the corral. The formula for the area of the corral, A = lw, shows area as a function of length and width. To write area as a function of width alone, we can substitute an expression for the length in terms of the width. Remember the formula for perimeter is P = l + w. Substitute the value of P and solve for l : Now substitute this expression for l into the area formula. A = lw b. Use b p to determine the x-coordinate of the vertex. a Use b q c to determine the y-coordinate of the vertex. 4a The vertex is located at (, ). The y-coordinate of the vertex represents the maximum area of the rectangle, which is. The x-coordinate of the vertex represents the width when this occurs, which is. c. Domain Range
Example 9: Quadratic Function Application On a typical day, the daily profit, P dollars, for a popcorn vendor is modelled by the equation P 60x 480x 10, where x is the price of a bag of popcorn. a. How much should the vendor charge for a bag of popcorn in order to maximize the daily profit? b. What is the maximum daily profit? Solution: