Graphs of Functions (Part 1) 2.5(1) Even/ odd functions Shifts and Scale Changes Even/ odd functions Shifts and Scale Changes

POD Simplify the following difference quotient, for f(x) = x 3. What does this expression become as h approaches 0? Simplify the following difference quotient, for f(x) = x 3. What does this expression become as h approaches 0?

POD Simplify the following difference quotient, for f(x) = x 3. What does this expression become as h approaches 0? Simplify the following difference quotient, for f(x) = x 3. What does this expression become as h approaches 0?

Even/ odd Functions Even functions are reflected over an axis-- which one? Odd functions are reflected over something else-- what? The acid test: For even functions, f(-x) = f(x) For odd functions, f(-x) = -f(x) (What does that mean in English?) Write which is which on the parent function handout. Even functions are reflected over an axis-- which one? Odd functions are reflected over something else-- what? The acid test: For even functions, f(-x) = f(x) For odd functions, f(-x) = -f(x) (What does that mean in English?) Write which is which on the parent function handout.

At each table, graph two of the following to see if they are even, odd, or neither As you graph each one, test it algebraically.

At each table, graph two of the following to see if they are even, odd, or neither Notice how p(-x) = p(x), once everything is simplified– it is an even function. The graph matches this.

At each table, graph two of the following to see if they are even, odd, or neither Notice how q(-x) = -q(x), once everything is simplified– it is an odd function. The graph matches this.

At each table, graph two of the following to see if they are even, odd, or neither Notice how k(-x) ≠ k(x) or -k(x), once everything is simplified– it is neither even nor odd. The graph matches this.

Vertical/ horizontal shifts Start by graphing y = x 2 and then y = x What do you notice about the relationship between the graphs? How does that compare to the relationship between the equations? Now, graph y = (x + 5) 2. How do the graph and equation compare to y = x 2 ? Start by graphing y = x 2 and then y = x What do you notice about the relationship between the graphs? How does that compare to the relationship between the equations? Now, graph y = (x + 5) 2. How do the graph and equation compare to y = x 2 ?

Vertical/ horizontal shifts y = x 2 y = x 2 + 5y = (x + 5) 2 (y – 5 = x 2 ) y = x 2 y = x 2 + 5y = (x + 5) 2 (y – 5 = x 2 )

Shifts and equations in general A vertical shift of c: y = f(x) + c y - c = f(x) A horizontal shift of c:y = f(x-c) How would each of these graphs compare to y = x 2 y = x 2 + 6x + 9 y = x 2 - 6x + 9 y = x 2 +3? A vertical shift of c: y = f(x) + c y - c = f(x) A horizontal shift of c:y = f(x-c) How would each of these graphs compare to y = x 2 y = x 2 + 6x + 9 y = x 2 - 6x + 9 y = x 2 +3?

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing) Start by graphing y = sin x Then, at tables graph one of these: y = 3sin x y = 1/3 sin (x) y = sin (3x)y = -3sin(x) What do you notice about the relationship between the graphs? Between the equations? Start by graphing y = sin x Then, at tables graph one of these: y = 3sin x y = 1/3 sin (x) y = sin (3x)y = -3sin(x) What do you notice about the relationship between the graphs? Between the equations?

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing) y = sin xy = 3sin x What do you notice about the relationship between the graphs? Between the equations? y = sin xy = 3sin x What do you notice about the relationship between the graphs? Between the equations?

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing) y = sin xy = sin (3x) What do you notice about the relationship between the graphs? Between the equations? y = sin xy = sin (3x) What do you notice about the relationship between the graphs? Between the equations?

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing) y = sin xy = (1/3)sin x What do you notice about the relationship between the graphs? Between the equations? y = sin xy = (1/3)sin x What do you notice about the relationship between the graphs? Between the equations?

Vertical/ horizontal scale changes (Vertical/ horizontal stretching and compressing) y = sin xy = -3sin x What do you notice about the relationship between the graphs? Between the equations? y = sin xy = -3sin x What do you notice about the relationship between the graphs? Between the equations?

Scale changes and equations in general A vertical stretch of c:y = cf(x) y/c = f(x) A horizontal stretch of c: y = f(x/c) We’re used to thinking of expansion with values of c greater than 1. How would we achieve a compression? A vertical stretch of c:y = cf(x) y/c = f(x) A horizontal stretch of c: y = f(x/c) We’re used to thinking of expansion with values of c greater than 1. How would we achieve a compression?

Scale changes and equations in general If c is negative, the graph will reflect over the y- axis when multiplied by x. reflect over the x-axis when multiplied by y. If c is negative, the graph will reflect over the y- axis when multiplied by x. reflect over the x-axis when multiplied by y.

The bottom line… If you change the equation, you do the opposite to the graph. When you change a graph, everything changes: intercepts, asymptotes, holes, domain, and range. If you change the equation, you do the opposite to the graph. When you change a graph, everything changes: intercepts, asymptotes, holes, domain, and range.

Scale change and translation Many graphs have a combination of scale change and translation (multiplication and addition). In that case, just like PEMDAS, you multiply then add– scale change then translation– in each direction.

Scale change and translation Try it with a point. Give the coordinates of the point (3, -1) after the function undergoes the transformation y = 2f(x+5) – 5. Remember, scale change first. Try it with a point. Give the coordinates of the point (3, -1) after the function undergoes the transformation y = 2f(x+5) – 5. Remember, scale change first.

Scale change and translation Try it with a point. Give the coordinates of the point (3, -1) after the function undergoes the transformation y = 2f(x+5) – 5. (-2, -7) Try it with a point. Give the coordinates of the point (3, -1) after the function undergoes the transformation y = 2f(x+5) – 5. (-2, -7)

Try it, if there’s time In groups, find a parent (tool kit) function that is even, or one that is odd. Graph it on your calculators. Shift it 3 units up and 4 units to the left. What is the new equation? Graph that to test your work. Stretch your original graph horizontally by a factor of 2, and reflect it over the x-axis. What is the new equation? Graph that to test your work. For a Take a Chance Award, come up to demonstrate your work. In groups, find a parent (tool kit) function that is even, or one that is odd. Graph it on your calculators. Shift it 3 units up and 4 units to the left. What is the new equation? Graph that to test your work. Stretch your original graph horizontally by a factor of 2, and reflect it over the x-axis. What is the new equation? Graph that to test your work. For a Take a Chance Award, come up to demonstrate your work.