Function Fundamentals
Can a function have more than one output value for a given input?
In the slope-intercept form of a line,
y = mx + b
What does the constant "b" represent? What about m?
b is the y-intercept (i.e. the y-value when x=0)
m is the slope of the line -- a.k.a the average rate of change!
f(x) = ax^2 + bx + c
let's say g(x) is a transformation of f(x) such that
g(x) = a f(x-h) + k
what do a, h, and k do to this function?
a stretches or compresses the function (if negative, reflects across the x-axis)
h shifts the function to the right or left (positive h means to the right)
k shifts the function up or down (positive k is up)
Factor the following expression:
X^2 - 6x -16
(x-8)(x+2)
Is the relationship between words and their definitions a function?
No! One word could have multiple definitions.
in the point-slope form of a line,
y-y0 = m(x-x0)
What are x0 and y0?
The x and y coordinates of a point that the line passes through
f(x)=a(x-h)^2+k
what is the vertex of the quadratic function given in this form?
(h, k)
Let the function r(x) be a straight, horizontal line with a y-intercept of 2. In other words, r(x) = 2.
how would you describe a new function s(x) = (1/2)r(x) ?
straight horizontal line with y-intercept of 1. I.e., new function s(x) = 1
let's say you have the following table:
x | 1 2 3 4 5 6
f(x) | 2 4 6 8 10 12
g(x) | 3 4 5 6 7 8
what is g•f(3)
g•f(3) = 8
g(f(3)) = g(6) = 8
what is the formula for average rate of change?
average rate of change = (f(b)-f(a))/(b-a)
In a random town, on average 2 people move to another place each year. Is the function representing the population of this town linear? Is the function increasing or decreasing?
The function is linear because the average rate of change is constant, and the function is decreasing.
if f(x) = 2(x-3)(x+5)
what is significant about the numbers 3 and -5?
what do we call this form?
3 and -5 are the x-values at which the value of the function is 0. In other words, where the function crosses/touches/intersects the x-axis.
It is called factored form
In what order would you perform the following transformations of f(x)?:
-(4f(x+3) - 10)
What is the quadratic formula?
x = (-b +- sqrt(b^2 - 4ac)) / 2a
Explain in words what the following expression means/what it is telling you to do: f•g(x)
Use g(x) (the output of the function g, evaluated at x) as an input for f(x)
Alternative answers:
- find g(x), plug that into f(x), compose
- f composed with g
y-4 = (-1/2)(x-6)
convert this to slope-intercept form!
y = (-1/2)x + 7
If a quadratic function is concave down, what can you say about the average rate of change?
It is negative
let f(2) = 4
If a new graph g(x) is defined as f(x-2), what is g(4)? (tip, graph this!)
g(4) = 4
since g(x) = f(x-2),
g(4) = f(4-2) = f(2) = 4
given g(x) = sqrt(x), what is the domain and range of this function?
domain: x = [0, infinity)
range: y = [0, infinity)
What's the difference between an abstract and model function for a given situation?
A model function has the same codomain as the abstract function, but its domain is restricted to make sense within the context of the model/problem
Let's say g(x) has an average rate of change = 4x. Is g(x) a linear function? why or why not?
No, g(x) is not linear because the average rate of change is not a constant.
Find the vertex of f(x) = 2x^2 - 6x + 7. Is it a minimum or a maximum?
(3/2, 5/2). It is a minimum because the coefficient of the squared term (2) is positive, so the function is concave up!
* will draw a graph on the board, ask to do some kind of transformation on it *
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for a quadratic function modelling an object being launched into the air, what does the vertex represent?
y-coordinate is maximum height reached, x-coordinate is time at which that height is reached (x-axis is often labelled with t instead in this case --> so the t-value is the time)