MATH 11011 EXPRESSING A QUADRATIC FUNCTION KSU IN STANDARD FORM Definitions: e Quadratic function: is a function that can be written in the form f(x) = aa? + br +.€ where a, b, and c are real numbers and a ¥ 0. e Parabola: The graph of a squaring function is called a parabola. It is a U-shaped graph. e Vertex of a parabola: The point on the parabola where the graph changes direction. It is the lowest point if a > 0, and it is the highest point if a < 0. Important Properties: e Standard form of a quadratic function: A quadratic function f(2) = ax? + bx + c can be expressed in the standard form by completing the square. e Once in standard form, the vertex is given by (h, k). e The parabola opens up if a > 0 and opens down if a < 0. Steps to put quadratic function in standard form: 1. Make sure coefficient on x? is 1. If the leading term is ax?, where a ¥ 1, then factor a out of each x term. 2. Next, take one-half the coefficient of x and square it. In other words, 1 2 (5 - coefficient of r) . 3. Add the result of step 2 inside the parenthesis. 4. In order not to change the problem you must subtract (a- result of step 2) outside the parenthesis. 5. Factor the polynomial in parenthesis as a perfect square and simplify any constants. Common Mistakes to Avoid: e When performing Step 4 above, do NOT forget to multiply the result of step 2 by the a that was factored out. 1. f(x) =a? + Quadratic function in standard form, page 2 PROBLEMS Express the quadratic function in standard form. Identify vertex. 4x —5 f(z) =(a@+2)?-9 Vertex = (—2, —9) 3. f(z) = —2* + 10x —2 Before we complete the square, we need to factor —1 from each x term. f(x) = —a? + 102 — 2 ==(9" 10x “jy =2 a ——a ae I f(z) = («&-3)?-8 Vertex = (3, —8) f(x) = —(a — 5)? +23 Vertex = (5, 23) Quadratic function in standard form, page 3 A. f(x) = 2a7+8e¢-1 6. f(x) = —4a? — 82 +3 Before we can complete the square, we need Before we can complete the square, we need to factor a 2 from each « term. to factor —4 from each x term. f(z) = 207 + 82-1 f(x) = 42? — 8243 =2(27+42 )-1 ==A(g? 427 \+3 1 2 1 2 (5 1) == A (5 2) =a gall 2 2 f(x) =2(2? +424 4)-—1- (2-4) f(x) = —4(a? + 22 +1) 4+3- (4-1) = 2(27 +42 +4)-1-8 = —A(a? + 29 +1) +3 — (4) =2(g4-9)7-9 = A(x? + 29 +1) +344 =-A(r+1)?+7 f(x) = 2(@ + 2)? -9 Vertex = (—2, —9) f(a) = —4(a@ +1)? +7 Vertex = (—1,7) 5. f(x) = 3x7 — 122 —10 Before we complete the square, we must fac- tor 3 out of each x term. f(x) = 3x? — 122 —10 = 3(27-4e )-—10 (5 ; -1) = (=2)? =4 f(x) = 3(x? — 42 +4) — 10 - (3-4) = 3(2? — 42 + 4) - 10-12 = 3 (0) 200 f(a) = 3(@= 2)" 92 Vertex = (2, —22) Quadratic function in standard form, page 4 7. f(x) =527 +5248 Before we can complete the square we need to factor 5 from each x term. f(x) = 5a? +5248 =5(2?+a2 )+8