2. ESSENTIAL UNDERSTANDING AND
OBJECTIVES
Essential Understanding: The graph of any
quadratic function is the transformation of the graph
of the parent function y = x2
Objectives:
Students will be able to:
Identify and graph quadratic functions
Identify and graph the transformations of quadratic
functions (reflect, stretch, compression, translation)
Solve for the minimum and maximum values of
parabolas
3. IOWA CORE CURRICULUM
Algebra
A.CED.1. Create equations and inequalities in one variable and use
them to solve problems. Include equations arising from linear and
quadratic functions, and simple rational and exponential functions.
Functions
F.IF.4. For a function that models a relationship between two quantities,
interpret key features of graphs and tables in terms of the quantities, and
sketch graphs showing key features given a verbal description of the
relationship.
F.IF.6. Calculate and interpret the average rate of change of a function
(presented symbolically or as a table) over a specified interval. Estimate
the rate of change from a graph.
F.IF.7. Graph functions expressed symbolically, and show features of the
graph, by hand in simple cases and using technology for more
complicated cases.
F.BF.3. Identify the effect on the graph of f(x) + k, kf(x), f(kx), and f(x+k)
for specific values of k (both positive and negative); find the value of k
given the graphs. Experiment with cases and illustrate an explanation of
the effects on the graph using technology.
4. VOCABULARY
Parabola: the graph of a quadratic function, it
makes a U shape
Quadratic Function: ax2 + bx + c
Vertex Form: f(x) = a(x – h)2 +k, where a doesn’t
equal zero, vertex is (h, k)
Axis of Symmetry: line that divides the parabola
into two mirror images. Equation x = h
Parent Function: y = x2
6. GRAPHING A QUADRATIC FUNCTION
Graphing a Function in the form f(x) = ax2
f(x) = (1/2)x2
Plot the vertex
Find and plot two points on one side of the axis of
symmetry
Plot the corresponding points on other side of the axis of
symmetry
Sketch the curve
Graph: f(x) = -(1/3)x2
What can you say about the graph of the function
f(x) = ax2 if a is a negative number?
7. TRANSFORMATIONS
Vertex form: f(x) = a(x-h)2 + k
Reflection: if a is positive the graph opens up, if a is
negative it reflects across the x-axis and opens
downward
If the parabola opens upward, the y coordinate of the
vertex is a minimum
If the parabola opens downward, the y coordinate of
the vertex is a maximum
Stretch a > 1 the graph becomes more narrow
Compression 0< a < 1 the graph becomes more flat
8. TRANSFORMATIONS
Standard form: f(x) = a(x-h)2 + k
Vertical Translation: k value, on the outside of the
parentheses. Moves graph up and down
Horizontal translation: opposite of the h value, on
the inside of the parentheses. Moves graph left and
right.
9. EXAMPLES
For the equations below, write the vertex, the axis
of symmetry, the max or min value, and the domain
and range. Then describe the transformations.
f(x) = x2 – 5
f(x) = (x – 4)2
f(x) = -(x + 1)2
f(x) = 3(x – 4)2 – 2
f(x) = -2(x +1)2
