Objectives
This lesson connects previous experience and knowledge of quadratic functions to a new way of writing quadratic
equations. Students will:
 understand what the parameters a, h, and k have to do with the graph of a parabola.
 write the vertex form of a parabola.
 sketch graphs of parabolas given the equation.
 write equations based on real-world problems using the vertex form.
 write their own application problems.
Essential Questions
 How can we determine if a real-world situation should be represented by a quadratic, polynomial, or
exponential function?
 How do quadratic equations and their graphs and/or tables help us interpret events that occur in the world
around us?
Vocabulary
 Parabola: In a Cartesian coordinate system, the graph that represents the general equation y = ax2 + bx + c
and is a conic section that is the intersection of a right circular cone and a plane parallel to a generating
straight line of that cone.
 Quadratic Function: A function of the form f(x) = ax2 + bx + c, where a, b, and c are real numbers.
 Standard Form: The form of a quadratic function expressed as an equation where a is the coefficient of x2,
b the coefficient of x, and c is the constant, commonly y = ax2 + bx + c.
 Vertex Form: The form of a quadratic equation where for functions of the form f(x) = ax2 + bx + c,
parameters a, h, and k in y = a(x − h)2 + k. determine characteristics of the parabola that represents the
function.
 Stretch Factor: The multiplicative factor by which a graph is compressed or stretched; i.e., a factor of 2
compress the graph, increase each y-coordinate by a factor of 2 relative to the original graph.
 Horizontal Shift: The number of units a graph is moved to the right or left.
 Vertical Shift: The number of units a graph is moved up or down.
 Maximum: A point, (a, f(a)) where f(a) is greater than all other y-values in the range.
 Minimum: A point, (a, f(a)) where f(a) is less than all other y-values in the range.
 Compressed: A graph that has a stretch factor greater than 1; compared to its parent function, the graph
looks as if it has been squeezed together, horizontally.
 Vertex of a Parabola: The point on a parabola (either a maximum or minimum, depending on the sign of a)
at which the parabola either stops increasing or stops decreasing.
 Transformation: Changing the shape of a parabola by multiplying or dividing a, reflections, or shifting it
horizontally and/or vertically. [IS.1 - Language Function]
Function Transformations / Translations (page 2 of 4)
Sections: Basic rules, Additional rules, Moving the points, Working backwards from the graph
The transformations so far follow these rules:






f(x) + a is f(x) shifted upward a units
f(x) – a is f(x) shifted downward a units
f(x + a) is f(x) shifted left a units
f(x – a) is f(x) shifted right a units
–f(x) is f(x) flipped upside down ("reflected about the x-axis")
f(–x) is the mirror of f(x) ("reflected about the y-axis")
There are two other transformations, but they're harder to "see" with any degree of accuracy. If you
compare the graphs of 2x2, x2, and ( 1/2 )x2, you'll see what I mean:
2x2
x2
_1/2 x2_
The parabola for 2x2 grows twice as fast as x2, so its graph is tall and skinny. On the other hand, the
parabola for the function ( 1/2 )x2 grows only half as fast, so its graph is short and fat. You can tell,
roughly speaking, that the first graph is multiplied by something bigger than 1 and that the third graph is
multiplied by something smaller than 1. But it is generally difficult to tell exactly what a graph has been
multiplied by, just by looking at the picture.
For instance, can you tell that the graph at right shows
1.4x2?
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Not likely.
The other more-difficult type of transformation is displayed below:
(2x)2
x2
( 1/2 x)2
As you can see, multiplying inside the function (inside the argument of the function) causes the graph to
get thinner or fatter. This looks a lot like the other multiplication transformation, and is about impossible to
identify from a graph. It helps to look at the zeroes of the graph (if it has more than one). For instance,
looking at y = x2 – 4, you can see that multiplying outside the function doesn't change the location of the
zeroes, but multiplying inside the function does:
2(x2 – 4)
x2 – 4
(2x)2 – 4
So the "left", "right", "up", "down", "flip", and "mirror" transformations are fairly straightforward, but the
"multiply" transformations, also called "stretching" and "squeezing", can get a little messy. Hope that they
aren't frequently required of you.
Typical homework problems on this topic ask you to graph the transformation of a function, given the
original function, or else ask you to figure out the transformation, given the comparative graphs.

Thinking of the graph of
x4, graph f(x – 2) + 1
f(x) =
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The graph of f(x) looks like this:
Looking at the expression for the
transformation, the "+1" outside
tells me that the graph is going to be moved up by one unit. And the "–2" inside the argument
tells me that the graph is going to be shifted two units RIGHT. (Remember that the left-right
shifting is backwards from what you might expect.)
Then my graph looks like this:
When they are having you graph by moving other graphs around, they can't be terribly critical of your
drawing, since you're not supposed to be making a T-chart and computing exact points. But do try to
make your graph look reasonable.
You can always "cheat", by the way, especially if you have a graphing calculator, by quickly graphing (x –
2)4 + 1 and verifying that it matches what you've drawn. But you do need to know how to do function
transformations, because there are ways to ask the questions that don't allow you to cheat....