Graphs curve sketching + factor theorem BARNES

Question 1

y = x^2

Answer 1

happy/sad face curve (negative or positive)
1 turning point
0, 1 or 2 real roots

Question 2

y = x^3

Answer 2

waterfall (if positive going up, if negative going down)
0 or 2 turning points
1, 2 or 3 real roots

Question 3

y = x^4
y = ax^4

Answer 3

flat happy face
or W shape (if negative M, upside down)
1 or 3 turning points
0, 1, 2, 3, or 4 real roots

Question 4

A factor (x - a) ⇒

Answer 4

crosses x-axis at a

Eg: y = (x - 2)(x + 2)

Question 5

A factor (x - a)^2 ⇒

Answer 5

touches x-axis at a

Eg: y = -(x + 3)^2

Question 6

A factor (x - a)3 ⇒

Answer 6

crosses x-axis & flattens at a

Eg: y = (x - 1)^3(x + 2)

Question 7

graph 1/x

Answer 7

Both have asymptotes at x = 0 and y = 0
like curly x on the side

Question 8

graph 1/x^2

Answer 8

Both have asymptotes at x = 0 and y = 0
1/x2 is always positive, and symmetrical

Question 9

y = x^1/2

Answer 9

increasing and getting less steep, starts at origin

Question 10

x^1/3

Answer 10

increasing and getting less steep
an S shape

Question 11

How do we find an inverse function geometrically?

Answer 11

y = x^2 and y = x^3, mirrored in y = x

x^1/2 stops at origin because with inverse functions like square roots, we restrict to one output

Question 12

Proportional means

Answer 12

two quantities are related through a constant, k

Question 13

y is directly proportional to x means

Answer 13

y = kx

Question 14

y is inversely proportional to x means

Answer 14

y = k/x

Question 15

factor theorem

Answer 15

If (ax - b) is a factor of the polynomial f(x) then f(b/a) = 0

Question 16

Using the factor theorem, show that (x + 1) is a factor of x^3 + 3x^2 - x - 3

Answer 16

if x+1 is a factor of fx, then f(-1) = 0

Question 17

Write x3 - x2 - 14x + 24 as a product of linear factors

Answer 17

Use trial and error to find the first factor, and algebraic division thereafter.
one factor is usually x, (x ± 1) or (x ± 2)

ans- (x-2) (x+4) (x-3)

Question 18

Two factors of q(x) = x3 + 4x2 + ax + b are (x - 1) and (x + 1).
Find the values of the constants a and b.

Answer 18

factor (x-1) goes to q(1)=0
factor (x-1) goes to q(-1)=0
solve simultaneously
b=-4
a=-1