web viewvertex and factorized form. vertex form: y=a x-h 2 +k . vertex(h,k)don’t forget to...

3-1 Factoring and solving Factoring methods( GCF, Sum/Product, DOTS, AC method) Solving Methods (Set equal to zero and Factor OR use Quad Formula OR Complete the

square.)

3-2 Discriminant b2−4 ac If ∆=0- roots are real, rational, equal If ∆>0 AND a perfect square- roots are real, rational, unequal. If ∆>0 AND not a perfect square- roots are real, irrational, unequal. If ∆<0 roots are imaginary.

3-3 Solving Quadratic Inequalities with the discriminant.1. Replace the inequality symbol with an equal sign and solve the resulting equation. The

solutions to the equation will allow you to establish intervals that will let you solve the inequality.

2. Plot the solutions on a number line creating the intervals for investigation. Pick a number from each interval and test it in the original inequality. If the result is true, that interval is a solution to the inequality.

DO THIS ONLY WHEN YOU END UP WITH AN INEQUALITY THAT HAS TWO SOLUTIONS (∆<0,∆>0)

3-4 Applications of quadraticsUse algebraic expressions to set up problems.

-Solve quadratics and ANSWER MUST MAKE SENSE.

3-5 Mixed Review- Make sure you look at answer key!

3-6 Vertex and factorized form. vertex form: y=a ( x−h )2+k vertex(h,k)

a. Don’t forget to change sign of only h coordinate when you put into equation.b. Complete the square when switching from standard form to Vertex form.

Factorized form: y=a(x−p)(x−q ). p and q are the roots.a.Don’t forget to change sign of BOTH roots when they go into equation

3-7 Equation from a graph. See what is given, choose the appropriate form, solve for a. turn back to standard form if asked to do so. to find X intercepts set y=0 and solve for x to find y intercepts set x=0 and solve for y

Practice Questions: Unit 3

Unit 3 – Quadratic Functions

1. a. Write the function f ( x )=2 x2−x−3 in the form f (x)=a(x−p)(x−q).

b. Sketch the graph of the function, labeling the x-intercepts and the y-intercepts.

2. The function f is given by f (x) = x2 – 6x + 13, for x 3.

(a) Write f (x) in the form (x – a)2 + b.

(b) Find the inverse function f –1.

3. The quadratic function f is defined by f (x) = 3x2 – 12x + 11.

(a) Write f in the form f (x) = 3(x – h)2 – k.

(b) The graph of f is translated 3 units in the positive x-direction and 5 units in the positive y-direction. Find the function g for the translated graph, giving your answer

in the form g (x) = 3(x – p)2 + q.

4. Consider: y=3 x2−15 x+9

a. Write the equation of f (x) in the form y=a ( x−h )2+k .

b. Hence, what are the coordinates of the vertex?

5.

6. The diagram shows part of the graph of y = a (x – h)2 + k. The graph has its vertex at P, and passes through the point A with coordinates (1, 0).

P

A– 1 0 1 x

y

2

1

(a) Write down the value of

(i) h;

(ii) k.

(b) Calculate the value of a.

7. (a) Express y = 2x2 – 12x + 23 in the form y = 2(x – c)2 + d.

The graph of y = x2 is transformed into the graph of y = 2x2 – 12x + 23 by the transformationsa vertical stretch with scale factor k followed bya horizontal translation of p units followed bya vertical translation of q units.

(b) Write down the value of

(i) k;

(ii) p;

(iii) q.

8. Let f ( x )=a ( x+3 )2−6.

a. Write down the coordinates of the vertex of the graph of f .

b. Given that f (1 )=2, find the value of a.

9. a. Express f ( x )=x 2−6 x+14 in the form f (x) = (x – h)2 + k, where h and k are to be determined.

b. Hence, or otherwise, write down the coordinates of the vertex of this parabola.

10.

11.

12.

13. For x2−2 x+m=0, find ∆ and hence find the values of m for which the equation has:

a. a repeated root

b. 2 distinct real roots

c. no real roots

14.

15.

16.

17.

18.

Unit 4 – Rational Functions

4-1 Pre-requisite skills. Inverse functions Graphing x= and Y= lines. Rational functions Ratio of two polynomials.

4-2 X and y intercepts on a function. To find the y-intercept of the graph of a function f , find f (0). To find the x-intercept(s) of the graph of a function f , set f ( x )=0. (Factor first!)

4-3 Asymptotes To find the vertical asymptotes of the graph of a function f , set the denominator of

the function equal to 0 and solve. For each value of x=c found, if f ( c )=non−zero ¿ ¿

0 then the graph of f has a vertical asymptote at x=c .

To find the horizontal asymptotes of the graph of a function f , use these guidelines.1. If the degree of the numerator is less than the degree of the denominator, then the graph of f has a horizontal asymptote at y=0.

2. If the degree of the numerator is equal to the degree of the denominator, then the graph of f has a horizontal asymptote given by the ratio of the leading coefficients.

3. If the degree of the numerator is greater than the degree of the denominator, then the graph of f does not have a horizontal asymptote.

Don’t forget the Rhyme! High up top nothing makes it stop High down low y equals zero.

4-4 Hole in the graph

If f ( c )=00 , then the graph of f has a hole at x=c . Any value of x that makes the

denominator equal zero should be excluded from the domain.

Procedure:1. Factorize top and bottom.2. Common factor equal to zero.3. Find other Vertical asymptotes.4. Check if the other vertical asymptote is also a hole.

DON’T FORGET! IF THERE IS AHOLE AT A PARTICULAR X VALUE, THERE CAN BE NO VERTICAL ASYMPTOTE OR X INTERCEPT THERE!

Practice Questions: Unit 4

1. Identify the following for the given function

domain _______________

vertical asymptote(s) ______________________

hole(s) ________________

horizontal asymptote ______________________

y-intercept _____________

x-intercept(s) _____________

2. Identify the following for the given function

domain _______________


hole(s) ________________



x-intercept(s) _____________

For each function given, find the following (a) domain (b) vertical asymptote(s) or hole(s), (c) horizontal asymptote, (c) y-intercept, (d) x-intercept(s).

3. f ( x )= x−23 x+2

domain _______________


holes ___________________



x-intercept(s) _____________

4. f ( x )= 1−xx2−x

domain _______________


holes ___________________



x-intercept(s) _____________

5. Let f (x) = 2x + 1 and g (x) = 3x2 – 4. Consider the function h ( x )= f (x )g (x) .

(a) Find the equations of any vertical asymptotes in the graph of h.

(b) Find the equations of any horizontal asymptotes in the graph of h.

(c) Are there any holes in the graph of h. Justify.

6. The diagram shows part of the graph of the

function f (x) = .

– pxq

The curve passes through the point A (3, 10). The line (CD) is an asymptote.

Find the value of

a) p;

b) q.

c) What is the domain of f (x)?

d) The graph of f (x) is transformed as shown in the following diagram. The point A is transformed to A’ (3, –10). Give a full geometric description of the transformation.

1 5

1 0

5

-5

-1 0

-1 5

C

A

D

y

x1 51 050– 5– 1 0– 1 5

web viewvertex and factorized form. vertex form: y=a x-h 2 +k . vertex(h,k)don’t forget to...

Documents