Day 6 Pre Calculus

Objectives Review Parent Functions and their key characteristics Identify shifts of parent functions and graph Write the equation of a transformed parent function

Let’s look at some “Parent Graphs” and their translations Constant Functions Linear Functions Quadratic Functions Cubing Functions Exponential Functions Logarithmic Functions Absolute Value Functions Square-root Functions

Translations – In General The transformation of any function f(x) looks like: where a represents the vertical stretch and orientation, h represents the horizontal shift, and k represents the vertical shift.

For each figure, you should be able to … Recognize the shape Give the parent graph equation Know their “locator” point (this helps with translations) Does it have asymptotes? (exponential and logarithmic) Translate the figure using the graphing form of the equation Give the domain and range of the parent or translated version

Constant Function Shape Parent graph Translation Domain Range Intercepts

Linear Function Shape Parent graph Translation Domain Range Intercepts

Linear Equations y = mx + b If you had to pick a “parent graph” what would it be for a linear equation? Thinking of “transformations of parent graphs”… What is another way we could think of “m” instead of slope? What is another way we could think of “b” instead of y-intercept The linear function is a line with a slope.

Quadratic Equation What is our parent graph? y=x 2 What is our locator point? Vertex at (0, 0) What are the intercepts of the parent graph? x- and y- intercept of (0, 0)

Quadratic function Let a, b, and c be real numbers a  0. The function: f (x) = ax 2 + bx + c or f(x) = a(x - h) 2 + k is called a quadratic function. The graph of a quadratic function is a parabola. Every parabola is symmetrical about a line called the axis (of symmetry). The intersection point of the parabola and the axis is called the vertex of the parabola. This is our “locator point” x y axis of symmetry f (x) = ax 2 + bx + c vertex

Quadratic Translations Parent graph y = a(x – h) 2 + k Vertex at (h, k) h is the horizontal shift k is the vertical shift a is the stretch or compression (shrink) and orientation

The Equation of a Parabola There are two forms for the equation of a parabola. General Form: y = ax 2 + bx + c where a, b, and c are real numbers and a  0 Standard Form: y = a(x – h) 2 + k also known as the graphing form Each form provides you with some information about the parabola

f (x) = x 2 Use the graph of f (x) = x 3 to graph h(x) = x g(x) = x 2 +3 Example: Horizontal Shifts

Quadratic Function Shape Parent graph Translation Domain Range

f (x) = x 3 The parent graph of a cubing function is f (x) = x 3. the point where the graph changes concavity is called the point of inflection – this is our “locator point”. x y Cubing Functions Point of inflection

The translation of the parent graph of a cubing function is the same as a quadratic. f (x) = a (x - h) 3 + k. h again is the horizontal shift k is the vertical shift a is the stretch or compression (shrink), using the point of inflection as the locator point. Cubing Functions Translating Cubing Functions

f (x) = x 3 Use the graph of f (x) = x 3 to graph h(x) = (x-2) 3. x y g(x) = (x – 2) 3 Example: Horizontal Shifts

Cubing Function Shape Parent graph Translation Domain Range

Exponential Functions The family of exponential functions are recognized by the exponent is the variable. y = a x, a > 0; a  1 The domain is all real numbers The range is all positive real numbers because a positive number raised to a power is positive It has an asymptote at y = 0

The graph of f(x) = a x, a > 1 y x (0, 1) Domain: (– ,  ) Range: (0,  ) Horizontal Asymptote y = 0 Graph of Exponential Function (a > 1) 4 4

Exponential Parent We will use y = 2 x as our parent graph The locator point is (0, 1)… why? For y = a x, a > 1 will pass through the point (0, 1) Why??? For all a ≠ 0, a 0 = 1

Transformations Given the equation: y = 2 (x-h) + k Vertical Shift y = 2 x + k Changes the horizontal asymptote – the new asymptote is y = k Changes the range – the new range is (k, ∞) Horizontal Shift y = 2 x-h What do you think would happen to y = -2 x ?

Use the graph of f (x) = 2 x to graph h(x) = 2 x - 4. (0, 1) Graph of Exponential Function (a > 1) (0,-3) y=0 y=-4

Exponential Function Shape Parent graph Translation Domain Range

Logarithmic Function It is the inverse of the exponential function Locator Point (1,0) Vertical Asymptote at x=0

Logarithmic Function Shape Parent graph Translation Domain Range

Graphing Absolute Values What are the domain and range of the parent graph? Intercepts? Any asymptotes? How would you describe the locator point? What’s the graphing equation? How would shifting affect the domain and range?

f (x) = x Use the graph of f (x) = x to graph g(x) = x + 3. g(x) = x +3 Example: Horizontal Shifts

Absolute Value Function Shape Parent graph Translation Domain Range

What would a square root function look like? Square Root Functions The graph is the upper branch of a parabola with vertex at (0, 0). The domain is [0,+∞) or {x: x ≥ 0}. The range is [0,+∞) or {y: y ≥ 0}. Example: Domain & Range x y (0, 0)

Square Root Functions Intercepts? Any asymptotes? How would you describe the locator point? What’s the graphing equation? How would shifting affect the domain and range?

Reflections Use the graph of y= to graph f(x) =. Example: Domain & Range (0, 0)

Square Root Function Shape Parent graph Translation Domain Range