y ax bx c y a x h 2 Math 11 Pre-Cal Quadratics Review

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1 Math 11 Pre-Cal Quadratics Review A quadratic function can e descried as y ax x c (or equivalent forms, see elow). There are an infinite numer of solutions (x,y pairs) to a quadratic function. If we plot these solutions, we get a paraola. A quadratic function has these properties: An unrestricted domain: x Restricted range, determined y the location of the vertex Its graph is symmetrical The axis of symmetry is determined y the location of the vertex A maximum or minimum location The graph will have 0, 1, or x-intercepts It has no slope as its rate of change is always changing What s useful aout the form Note: All three forms tell us the direction of the opening. That comes from the sign of a. The leading coefficient a is the same in all three forms. This a coefficient also controls how wide or narrow the paraola is larger a s = narrower, while smaller a s = wider. Moving etween forms Factored Form Standard * Form Vertex * Form y a( x r)( x s) y ax x c y ax h k Very easy to solve 0 a( x r)( x s) Therefore, it tells us the roots (x-intercepts): r and s Tells us the y-intercept: the value of c Also: easy to get to factored form (if possile) and partially factored form Only form for which you can use the quadratic formula The value affects the position of the vertex x ) ( vertex a Factor, if possile Complete the square Tells us the vertex (h, k). Also: easy(ish) to solve, since there is only an x in one place we can use standard algeraic solving. Expand, simplify Transformations of quadratic graphs: y ax h k in vertex form. p is a horizontal translation q is a vertical translation a is the vertical stretch factor. - We can use these transformations to graph paraolas. Expand, simplify Sketching the graph of a quadratic equation without graphing technology: Method 1: Making a tale of values and plotting (poor method) Method : Plot using three points: two symmetrical points on the curve and the vertex. Method 3: Plotting the vertex (calculated or read from the equation in vertex form), the y-intercept and a symmetrical point to the y-intercept (you can also plot the two x-intercepts with the vertex instead) Method : Plotting the vertex, and using the over/up pattern as dictated y the vertical stretch in the paraola.

2 Solving quadratic equations to find 0, 1, or solutions: (slight differences etween roots, x-intercepts and zeros) Method 1: With a graphing calculator determine intersection points. Method : By factoring and using the zero product property. Method 3: The quadratic formula. If ax x c 0 then x Method : Complete the square this is new this semester, (not an algeraic solution) review the idea closely. Each have their advantages/disadvantages, and you should e proficient with all four. The imaginary numer, complex roots and the discriminant ac : Some quadratics have no x-intercepts, which results in zeros/roots that aren t real numers. These situations require us to use the imaginary numer, i ( i 1, ). Example: Let s see an example. Solve f(x) = x +x+3: x 9 0 solves as x 9, so x 3 i. f ( x) x x 3 is not factorale. Fill into the quadratic formula and simplify to get: x. Use radicals knowledge to simplify more: Now, we use the imaginary numer: x 1, then divide out the common factor: x, ut since i 1, we get: i x 8 x. as roots/solutions. Since the numer under the root controls whether the function has complex roots or real roots, mathematicians named the numer under the root. It s called the discriminant (D). The value of the discriminant is a shortcut to tell us the nature of the roots. How can we summarize the x-intercepts/roots/zeros situations: Graph Example(s) Nature of Roots Quadratic formula situation Discriminant (D) situation No x-intercepts, so Graphs that never cross the x axis there are two complex end up with negative values under If a function has two roots we will use i to the root in the quadratic formula complex roots, that function descrie them. (that s why they have complex has ac 0 numers for the roots). No x-intercepts, so roots are not visile on graph. e.g. i x aove example from Two x-intercepts, so there are two different, real roots. e.g. f ( x) x x 3ends up 8 with x in the quadratic formula. ( complex roots) Graphs that cross the x axis twice have positive values under the root in the quadratic formula, so their roots are real numers. e.g f ( x) x x 3ends up with x 0 ac a in quadratic formula. (so different roots). Conversely, ac 0 means that a function has two complex roots. If a function has two different, real roots, that function has ac 0 Conversely, ac 0 means that function has two different, real roots. One x-intercept, so there are two, equal, real roots. Graphs that sit on the x axis have ZERO under the root in the quadratic formula, so their two roots are the same numer. e.g. y = x -1x+18 ends up with 1 0 x in the quad. formula, so x = 3 is the only root/solution. If a function has two equal real roots, that function has ac 0 Conversely, ac 0a means that function has two equal real roots.

3 Quadratics practice questions f x x 1x. 1. A quadratic function is given y 3 5 a. Write out domain and range of function.. Write the function in vertex form (use two different methods to do this) c. Use the vertex form to descrie the transformations present in the function. d. Sketch a good graph of the function, laeling vertex, x-intercepts, y-intercepts and one other point.. The function h t.9t 15.7t 1 descries the height of a volleyall in metres, h, as a function of time, t, in seconds after it was hit. a. From what height was the all hit?. After how many seconds does the all land? c. After how many seconds does the all reach its maximum height? d. What is the all s maximum height? e. For how many seconds was the all at or aove a height of 7m? f. State the domain and range of the function with regards to the word prolem. 3. Factor: a) (x ) + 7(x ) + 5 ) x 3 6x c) x x d) 8x + 33x + 5. The sum of numers is 80 and the sum of their squares is a minimum. Find the numers. 5. A paraola has a vertex of (1, -3) and passes through (, 15). Find the equation of the quadratic function. 6. a. Find the zero(es) of each y factoring: y 6x x y 10x 13x 3. Find the x-intercepts y completing the square: y x x y x x c. Find the roots y the quadratic formula: x 7x 3 0 3x 1x Find the value of d so the equation x dx d 1 has: a. two equal real roots. two different, real roots c. two complex roots. 8. The hypotenuse of a right triangle is 3 cm. If the other two sides add up to 6 cm, find their lengths. 9. Given the paraola at the right, determine its equation in oth vertex and standard form. Use the equation to determine its x-intercepts. 10. A square swimming pool with a side measuring 16 m is to e surrounded y a ruerized floor covering of uniform width. If the area of the floor covering equals the area of the pool, find the width of the ruerized covering. 6, A farmer is uilding a fenced in garden along the side of their arn. Only three sides of the garden will need fencing. The farmer has 300 meters of fencing, determine the dimensions that will maximize the area. 1. Two numers have a sum of 15 and a product that is a maximum. Determine the numers. 13. Determine two quadratic equations with roots of 3 and Explain why the function y ( x 3) 6 will have two distinct x intercepts. 15. Determine the equations of one paraola with a range of y y 3, y R

4 Solutions for Quadratics: 1. x R y y 3, y R ) y ( x 8x 16 16) 3 5 y ( x ) ( 16) y ( x ) 3 3 You could have also used x a to find x part of vertex, then got y part of vertex from original equation and plugged into vertex form. c) R x, VS of 3/, HT of and VT of 3/ d) 5 y x Points to lael: a) 1m (t = 0) ) 0.9t 15.7t 1 t 0.06 t 3.7 (QF or calculator) Hits the ground in 3.7s. c) 15.7 t 1.60 s d) h(1.60) 13.58m (.9) e) 7.9t 15t t 15t 6 t 0. t.76 So aove 7 m for, =.3 seconds f) Domain: t 0 t 3.67, t R 3. a) Let a x a 7a5 (a5)( a1) 1 Range:h( t) 0 h( t) , h(t) R (( x ) 5)( x 1) (x1)( x1) c) x ) x( x )( x ) ( x ) d) (8x1)( x ) Let the numers e r and s and the let the minimum value e m (like a y value) r s 80 s 80 r m r s m r (80 r) m r r r m r r To get min need vertex : r 0 () Therefore the r value at the minimum is 0, then s would e 80 0 = 0 as well. And the minimum value of the sum of their squares is m Think aout this, there are many cominations of numers that add to 80, ut of all the cominations 300 is the smallest sum of their squares.

5 5. Vertex (1, -3) and point (, 15) Use the vertex form of the quadratic: y a x fill in po ( 1) 3 int (,15) 15 a( 1) a a y ( x 1) 3 6. a) 6x x 0 (3x )(x1) 0 x x x 13x 3 0 (x3)(5x11) 0 x x ) ( x 5 x ) x 8 0 ( x ) ( x ) x x x 8 x 3 ( x x ) x 0 ( x ) ( x ) x x x i 5 7i For the questions aove you most certainly could have taken a few less steps when I did these in class I usually worked on oth sides of the equal sign right from the start.. c) x () x 7 7 ()( 3) 7 73 Divide through y 3: x (1) (1)( 1) 100 x 10 x x 7 x 3 7. x dx d 1 You need to use the discriminant for these. Two Equal Roots: Two Different Real Roots: Two Complex Roots: d d ac 0 (1)( d) 0 d 0 dd ( ) 0 d 0 d ac 0 dd ( ) 0 d 0or d To determine where the discriminant is greater than zero or less than zero you can do sign analysis. ac 0 dd ( ) 0 0d

6 8. Let the legs e x, 6 x and the hypotenuse e 3. Now use Pythagorean Theorem: So the legs could e 16 and 30 or 30 and y a x fill in po ( ) 18 int (6, 1) 1 a(6 ) 18 6 a 3 3 a y ( x ) 18 y 3 ( x ) 18 y x x 3 ( 8 16) 18 3 y x x x 1x 6 ( QF) x0.5 x7.6 x (6 x) 3 x 9x ( x16)( x 30) 0 x16 x Let x e the width of the strip. If the area of the strip is equal to the area of the small square, then the larger square will have an area of twice the smaller square. Area small Square = 16 x 16 =56 Area of large square = x 56 = x + 16 (x16)(x16) 51 x 6x x 6x 56 0 x x QF ( ) x The width of the strip is 3.1 meters. 11. To get coordinates of the vertex: 1. r s 15 r 15 s 13. M r s M (15 s)( s) M s 15s s 15 ( 1) r x x 3 1 x 3 x 1 x 3 0 x 1 0 y a(x 3)(x 1) where a 0 1. The paraola has a positive y intercept and concave down. (reflection) It will definitely cross the x axis twice. 15. y a x p a ( ) 3, 0