4-1 QUADRATIC FUNCTIONS AND TRANSFORMATIONSChapter 4 Quadratic Functions and Equations

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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

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, k f(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.

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

QUADRATIC FUNCTION

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?

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

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.

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

HOMEWORK

Pg. 199 – 200 # 9-33 (3s) 35-37, 38, 40 – 42