Quadratic function and equation #17 - Questions and Answers

Question 1 What is the degree of a quadratic function f? What is the standard form of a quadratic function? How do you put a quadratic function into standard form? Answer A quadratic function is of the form f(x) = ax² + bx + c So, the degree of quadratic function is = 2 The standard form of a quadratic function is f(x) = ax² + bx + c. If a quadratic function is not in the standard form, then we try to put it in f(x) = ax² + bx + c. form to get the standard form. For example: f(x) = 2(x - 1)² + 3 f(x) = 2(x² – 2x + 1) + 3 f(x) = 2x² - 4x + 2 + 3 f(x) = 2x² - 4x + 5 (standard form) Question 2 Express x² + 8x in the form (x + a)² + b Answer 1 To express x² + 8x in the form (x + a)² + b, we need to complete the square. For an expression of the form x² + Bx, we can complete the square by adding and subtracting (B/2)². You need to add and subtract this number so the new expression will be equivalent to the original expression. For x² + 8x, B = 8 so we need to add and subtract (8/2)² = 4² = 16. This gives: x² + 8x = x² + 8x + 16 - 16 Group the first three terms to form a perfect square trinomial: = (x² + 8x + 16) – 16 A perfect square trinomial of the form x² + Bx + (B/2)² factors to (x + B/2)²(x + 2B)². Since B/2 = 8/2 = 4, then x² + 8x + 16 factors to (x + 4)². Therefore: (x² + 8x + 16) - 16 = (x + 4)² – 16 Answer 2 Step 1: Let's begin by adding and subtracting (8/2)² to the expression: x² + 8x + (8/2)² - (8/2)² Simplifying this expression, we have: x² + 8x + 16 - 16 Step 2: Now, we can group the first three terms together and rewrite the expression as follows: (x² + 8x + 16) - 16 The first three terms inside the parentheses form a perfect square trinomial, which can be factored as (x + 4)². Therefore, we have: (x² + 8x + 16) - 16 = (x + 4)² - 16 Simplifying further, we get: (x + 4)² - 16 Now the expression x² + 8x has been expressed in the form (x + a)² + b, where a = 4 and b = -16. Answer 3 To express x² + 8x in the form (x + a)² + b, we need to complete the square. First, let's add and subtract the square of half the coefficient of x (which is 4) within the expression: x² + 8x = (x² + 8x + 16) - 16 Now, we can rewrite the expression as a perfect square: x² + 8x = (x² + 8x + 16) - 16 = (x + 4)² - 16 Therefore, the expression x² + 8x can be expressed in the form (x + 4)² - 16. Question 3 Suppose that the combined area of two squares is 360 square feet. Each side of the larger square is three times as long as a side of the smaller square. How big is each square? Answer Let x be the side length of the larger square and y be the side length of the smaller square. The area of a square is s² where s is the side length so the larger square has an area of x² and the smaller square has an area of y². The combined area is then x² + y². The combined area is 360 square feet - x² + y² = 360. If the larger square's side is three times as long as the smaller square's side, then x=3y. Substitute x=3y into x² + y² = 360 x² + y² = 360 (3y)² + y² = 360 9y² + y² = 360 10y² = 360 y² = 36 √y² = ± √36 y = ± 6 Since the sides of the squares must be positive, then y=6 so x = 3y = 3(6) = 18. The larger square then has side lengths of 18 ft and the smaller square has side lengths of 6 ft. Question 4 Find the vertex, the axis of symmetry, and the y-intercept of the graph. Answer The vertex of a quadratic function is the lowest point when it opens upward or the highest point when it opens downward. The given graph opens upward so we need to look for the lowest point in the graph. The lowest point is at (2,-1) so the vertex is (2,-1). The axis of symmetry is the vertical line passing through the vertex. If a quadratic function has a vertex of (h,k), then the axis of symmetry is x=h. Since the vertex of the graph is (2,-1), then the axis of symmetry is x=2. The y-intercept is where the graph intersects the y-axis. The graph intersects the y-axis at (0,1). The y-intercept is then 1. Question 5 Identify the vertex, complete the table and graph g(x) = (x - 4)² – 5. Section 6 - Topic 5 Graphing Quadratic Functions Using Vertex Form - Part 1 Identify the vertex, complete the table and graph g(x) = (x − 4)² – 5. x | g(x) Vertex: Answer The vertex form of a quadratic equation is y = a(x - h)² + k where (h,k) is the vertex. Comparing y = a(x - h)² + k and g(x) = (x - 4)² - 5 gives h=4 and k=-5. The vertex is then (h,k) = (4,-5). Replace the values of x on one side of the vertex into g(x) to find the corresponding y- coordinates and complete the table: x | g(x) 5 | (5-4)² - 5 = 1 - 5 = -4 6 | (6-4)² - 5 = 4 - 5 = -1 7 | (7-4)² - 5 = 9 - 5 = 4 8 | (8-4)² - 5 = 16 - 5 = 11 Plot the vertex and the four points from your table. A quadratic is symmetric about its vertex so plot the mirror images of the four points on the other side of the vertex: