Chapter 4 Sketching Polynomials

Question 1

Sketching quadratics

Answer 1

Procedure:

1. Find axis of symmetry using x = -b/2a formula then plug in x value in equation to find y
2. Use ax^2 + bx + c (USE C) to find the y intercept
3. Solve the quadratic to find roots
4. Plot everything
5. Table of values may be used

Question 2

Sketching cubic equations Pt.1

Answer 2

Use rational root theorem to find roots

When x = 0 find y, gives you origin

Question 3

Sketching cubic equations Pt.2

Answer 3

Differentiate, then let the quadratic = 0

Solve quadratic and plug into original equation pre-differentiation

Gives you turning points

Plot all the points found which includes turning points, x-intercept and y-intercept, origin

Question 4

Sketching cubic equations in the form ax^3 + bx^2 +cx + d or (x+a)(x+b)(x+c)

Answer 4

1. Find roots
2. Substitute x = 0 in the factorised form to find y-intercept
3. Plot them you can use table of values for more accurate answers

Question 5

Sketching cubic equations in the form y = x^3

Answer 5

Question 6

Sketching the reciprocal function in the form y = k/x where k is constant

Answer 6

Question 7

Finding Horizontal and Vertical Asymptotes and the intercepts and the slant

Answer 7

1. To find x/vertical asymptote, you will take the denominator and equal it to 0 and find x
2. To find y/horizontal asymptote it depends on 3 conditions
   
   1. Consider the degree of the leading term, if the denominator’s degree is greater than the numerator then the horizontal asymptote will be y = 0
   2. Considering the same thing, if the numerator’s degree is equal to the denominator
      then the horizontal asymptote will be the lead coefficients divided by each other
   3. Consider the same thing, if the numerator’s degree is larger than the denominator
      then there will be no horizontal asymptote it would be a slant
3. Make numerator equal to 0 to find the x intercept
4. Make all x values 0 then solve to find the y intercept or divide the constants
5. To find slant do synthetic division/algebraic division, slant is only finable when the degree of numerator is 1 more than denominator
6. If you have a quadratic, you factor them and cancel the common factors in numerator and denominator the cancelled factor is a hole

Question 8

Transformations on curves

Answer 8

Question 9

To sketch an exponential graph

Answer 9

1. Find the x values by making the exponent equal to 0 and 1
2. Plug in the corresponding x values into the equation and find y values
3. if the equation has a constant value then that will be the horizontal asymptote
4. Way to rmb direction is in the first picture