Math 1330, Chapter 2, Section 1
Linear functions
Definition:f(x) = y = mx + bwhere m and b are real numbers
and m 0.
Now, we all know the basic truth:
Two points determine a line.
Let’s expand this to:Two facts determine a line.
Review:
slope, m
m = 0 horizontalm > 0 increasing m < 0 decreasing
x-intercept and y-intercept: Box!
- Tell me everything about .
Domain, range, incr/decr, intercepts.
point slope formula
Solve for a formula for the inverse function:
Problems
1.Suppose the point ( 1, 2) is on the graph of a linear function and the point ( 4, 3) is on the inverse function’s graph. What is the formula for the function?
More facts:Parallel lines have the same slope and
perpendicular lines have slopes that multiply to 1
(ie, )
2.Find an equation for the line that is perpendicular to 4x 2y = 5 and contains the point ( 4, 1).
3.Given, find f(x).
Quadratic functions
old way:With A, B, and C real numbers and
A 0.
preferred:h, k, and A are real numbers and
A 0.
Parts of a quadratic:
Vertex:( h, k)
h is the number to the right of the minus sign,
k is the number to the right of the plus sign
k is a maximum value [down, A<0, negative: frown; range is [, k]]
sketch:
or a minimum value [up, A>0, positive: smile; range is [k,]]
sketch:
Axis of symmetry:x = h
x intercepts:none, one*, or two.
Solve the equation for x when y = 0.
Use factoring or the (h, k) form whichever is easier
*”tangent” to the x axis
y intercept or ( 0, C). Solve the equation for y when x = 0.
shape factor:AIf A = 1, the parabola is the same shape as .
If , the parabola is narrower than .
If , the parabola is wider than .
mnemonic: Fractions make it Fatter.
Find all of the above for
Shape factor:What is the shape of the parabola?
might need to put in homemade minus and plus signs to see h and k
Vertex:k is max or min?
Axis of symmetry
y intercept
x interceptstwo ways to do this:
1. CTS form and y = 0[nb: if this gives imaginary numbers, no x intercepts]
2. factoring
sketch of graph:
Now, what about ?
Shape factor:
y intercept:
x intercepts:
Do Complete the Square to get the vertex and shifting info:
[See the handout on my website for Completing the Square to get more practice on these]
take b, divide it by two and square it
add this to the left hand side in between x and 6, then add appropriately to rebalance the equation
turn the first 3 terms into a perfect square and combine the last two
pick out h and k
Vertex:vertex y is max or min?
Axis of symmetry:
Sketch:
Let’s do one with a shape factor other than 1.
Find everything about
Shape factor:what is the shape?
y intercept:
x intercepts:none…the work gives irrational numbers
Complete the square
factor the 2 out of the first two terms, write them in parentheses with space:
take the “inside” b, divide it by 2, square it and add it into the parenthesis
add appropriately to the third term to rebalance the equation
finish writing it up showing the perfect square:
Vertex and axis of symmetry:
vertex y is max or min
Sketch:
Another one:
This one has a negative shape factor.
Shape factor:1
y intercept
x intercepts
Complete the square:
Factor the shape factor out of the first two terms…leave a space:
take the inside b, divide by 2, square it, add it inside.
add appropriately to the third term to rebalance the equation
complete the square:
Vertex and axis of symmetry:max or min
Sketch the graph
Working with and g(x) = x + 1. Tell me everything about
result of the composition:
Shape factor:
shifting instructions – do Complete the Square to get these
vertex: is k a max ( ) or min value ( )
axis of symmetry
y intercept
x intercepts
go to results of composition
use CTS
graph
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