Chapter 3 Review Pre-Calculus

Determine what each graph is symmetric with respect to y-axis y-axis, x-axis, origin, y = x, and y = -x y-axis, x-axis, and origin

The graph of each equation is symmetric with respect to what? Two squared terms, with same coefficients means it is an circle with center (0, 0) Symmetric with respect to x-axis, y-axis, origin, y = x, and y = -x Two squared terms, but different coefficients means it is an ellipse with center (0, 0) Symmetric with respect to x-axis, y- axis, and origin One squared term means it is a parabola shifted up 5 units and more narrow. Symmetric with respect to the y-axis

Graph each equation:

Determine whether each function is even, odd or neither. If all the signs are the same, then the function is EVEN Figure out f(-x) and –f(x)

Determine whether each function is even, odd or neither. If all the signs are opposite and the same, then the function is NEITHER even or odd. Figure out f(-x) and –f(x)

Determine whether each function is even, odd or neither. If all the signs are opposite then it’s ODD Figure out f(-x) and –f(x)

Describe the transformation that relates the graph of to the parent graph THREE UNITS TO THE LEFT Describe the transformation that relates the graph of to the parent graph THREE UNITS UP, AND MORE NARROW Describe the transformation that relates the graph of to the parent graph FOUR UNITS TO THE RIGHT, AND THREE UNITS UP

Describe the transformations that has taken place in each family graph. Right 5 units Up 3 units More Narrow More Narrow, and left 2 units

Describe the transformations that has taken place in each family graph. More Wide, and right 4 units Right 3 units, and up 10 units More Narrow Reflected over x-axis, and moved right 5 units

Describe the transformations that has taken place in each family graph. Reflect over x-axis, and up 2 units Reflected over y-axis Right 2 units

FINDING INVERSE FUNCTIONS STEPS Replace f (x) with y Interchange the roles of x and y Solve for y Replace y with f -1 (x) Find the inverse of,

FINDING INVERSE FUNCTIONS STEPS Replace f (x) with y Interchange the roles of x and y Solve for y Replace y with f -1 (x) Find the inverse of f (x) = 4x + 5

STEPS Replace f (x) with y Interchange the roles of x and y Solve for y Replace y with f -1 (x) Find the inverse of f (x) = 2x 3 - 1

STEPS Replace f (x) with y Interchange the roles of x and y Solve for y Replace y with f -1 (x) Find the inverse of

Steps for finding an inverse. 1.solve for x 2.exchange x ’ s and y ’ s 3.replace y with f -1

Graph then function and it’s inverse of the same graph. Parabola shifted 4 units left, and 1 unit down Now to graph the inverse, just take each point and switch the x and y value and graph the new points. Ex: (-4, -1) becomes (- 1, -4) Finally CHECK yourself by sketching the line y = x and make sure your graphs are symmetric with that line.

Graph then function and it’s inverse of the same graph. Cubic graph shifted 5 units to the left Now to graph the inverse, just take each point and switch the x and y value and graph the new points. Ex: (-5, 0) becomes (0, -5) Finally CHECK yourself by sketching the line y = x and make sure your graphs are symmetric with that line.

Graph then function and it’s inverse of the same graph. Parabola shifted down 2 units Now to graph the inverse, just take each point and switch the x and y value and graph the new points. Ex: (0, -2) becomes (-2, 0) Finally CHECK yourself by sketching the line y = x and make sure your graphs are symmetric with that line.

Vertical Asymptotes: Horizontal Asymptotes: Holes: Intercepts:

Determine if each parabola has a maximum value or a minimum value. y = ax 2 + bx + c “a” is positive so that means it opens up, and has a minimum “a” is negative so that means it opens down, and has a maximum

Graph each inequality:

Find the maximum point of the graph of each: Plug -1 into the original function Max (-1, 9) Since we know it is a parabola opening down, it definitely has a maximum 4x = 0 x = 0 x + 2 = 0 x = -2 x – 2 = 0 x = 2 (0, 16) (-2, 0) (2, 0) Test around each point and find that (0, 16) is a max

Find the x and y intercepts of each: x – intercept: y = 0 0 = x x = (x + 4)(x + 8) x + 4 = 0 x = -4 (-4, 0) x + 8 = 0 x = -8 (-8, 0) y – intercept: x = 0 (0) (0) + 32 (0, 32) x – intercept: y = 0 0 = x (x 2 – 12x + 35) 0 = x (x - 7)(x - 5) x = 0 (0, 0) x - 7 = 0 x = 7 (7, 0) x - 5 = 0 x = 5 (5, 0) y – intercept: x = 0 (0) (0) (0) (0, 0)

Without graphing, describe the end behavior of the graph of Positive coefficient, even power means it rises right and left Negative coefficient, even power means it falls right and left positive coefficient, odd power means it rises right and falls left

Without graphing, describe the end behavior of the graph of Positive coefficient, even power means it rises to left and falls to right Positive coefficient, odd power means it rises right and falls left positive coefficient, even power means it rises right and rises left

Determine whether each function is even, odd, or neither. Neither Neither – I know it has the same signs but it is just the top half of a parabola on it’s side. ODD

Graph the function, Find the inverse equation, Graph the inverse on the same graph. Is the inverse a function? The inverse is not a function

Determine the asymptotes for the rational function, then graph it VA: x = 3 HA: y = 0 Graph your asymptotes first and then test points left and right of each vertical asymptote.

Graph the inequality

Find the derivative of the function:

Find the x and y intercepts of each: x – intercept: y = 0 0 = x x = (x + 4)(x + 8) x + 4 = 0 x = -4 (-4, 0) x + 8 = 0 x = -8 (-8, 0) y – intercept: x = 0 (0) (0) + 32 (0, 32) x – intercept: y = 0 0 = x (x 2 – 12x + 35) 0 = x (x - 7)(x - 5) x = 0 (0, 0) x - 7 = 0 x = 7 (7, 0) x - 5 = 0 x = 5 (5, 0) y – intercept: x = 0 (0) (0) (0) (0, 0)

Ex. Find the critical points of f(x) = 3x 4 – 4x 3 on the interval[-1, 2]. f’(x) = 12x 3 – 12x 2 0 = 12x 3 – 12x 2 0 = 12x 2 (x – 1) x = 0, 1 are the critical numbers Evaluate f at the endpoints of [-1, 2] and the critical #’s f(-1) = f(0) = f(1) = f(2) = min. max.

Ex. Find the critical points of f(x) = 2x – 3x 2/3 on [-1, 3] = 0 C. N.‘s are x = 1 because f’(1) = 0 and x = 0 because f’ is undefined f(-1) = f(0) = f(1) = f(3) = min. max. Evaluate f at the endpoints of [-1, 3] and the critical #’s.

Sketch the graph, then describe the graph.