Company LOGO Review Pre-Calculus

Determine whether the graph of each relation is symmetric with respect to the x-axis, the y-axis, the line y = x, the line y = -x, and/or the origin. Find a point that works: (9, 2) Now test to see if each point exists according to the chart: (a, b) y-axis(-a, b) x-axis(a, -b) origin(-a, -b) y = x(b, a) y = -x(-b, -a) Does (-9, 2) exists? NO Does (9, -2) exists? YES Does (-9, -2) exists? NO Does (2, 9) exists? NO Does (-2, -9) exists? NO So this graph is symmetric w/ respect to the x-axis

Determine whether the graph of each relation is symmetric with respect to the x-axis, the y-axis, the line y = x, the line y = -x, and/or the origin. I know this is a ellipse because it has two squared terms with two different coefficients. It has a center (0, 0) So this graph is symmetric w/ respect to the x-axis, y-axis, and origin.

Find a point that works: (1, 5) Now test to see if each point exists according to the chart: (a, b) y-axis(-a, b) x-axis(a, -b) origin(-a, -b) y = x(b, a) y = -x(-b, -a) Does (-1, 5) exists? NO Does (1, -5) exists? NO Does (-1, -5) exists? YES Does (5, 1) exists? NO Does (-5, -1) exists? NO So this graph is symmetric w/ respect to the origin Determine whether the graph of each relation is symmetric with respect to the x-axis, the y-axis, the line y = x, the line y = -x, and/or the origin.

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

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

Let’s consider the function and compute some values and graph them. x f (x) Is this a function? Yes What will “undo” a cube? A cube root This means “inverse function” x f -1 (x) Let’s take the values we got out of the function and put them into the inverse function and plot them These functions are reflections of each other about the line y = x (2,8) (8,2) (-8,-2) (-2,-8)

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.

Graph Vert asymp: x 2 -4=0 x 2 =4 x=2 & x=-2 Horiz asymp: (degrees are =) y= 3 / 1 or y=3 x y left of x=-2 asymp. Between the 2 asymp. right of x=2 asymp.

Domain: all real #’s except -2 & 2 Range: all real #’s except 0<y<3

Find the horizontal asymptote: Exponents are the same; divide the coefficients Bigger on Top; None Bigger on Bottom; y=0

Find the domain. Excluded values are where your vertical asymptotes are

Find horizontal or oblique asymptote by comparing degrees degree of the top = 0 remember x 0 = 1 degree of the bottom = 2 If the degree of the top is less than the degree of the bottom the x axis is a horizontal asymptote.

Find some points on either side of each vertical asymptote xR(x)R(x) Choose an x on the left side of the vertical asymptote Choose an x in between the vertical asymptotes. Choose an x on the right side of the vertical asymptote. 41

Connect points and head towards asymptotes. There should be a piece of the graph on each side of the vertical asymptotes. Pass through the point and head towards asymptotes Pass through the points and head towards asymptotes. Can’t go up or it would cross the x axis and there are no x intercepts there. Pass through the point and head towards asymptotes Go to a function grapher or your graphing calculator and see how we did.

Find the domain. Excluded values are where your vertical asymptotes are. Let's try another with a bit of a "twist": But notice that the top of the fraction will factor and the fraction can then be reduced. We will not then have a vertical asymptote at x = -3, It will be a HOLE at x = -3 vertical asymptote from this factor only since other factor cancelled.

Find horizontal or oblique asymptote by comparing degrees degree of the top = 1 degree of the bottom = 1 If the degree of the top equals the degree of the bottom then there is a horizontal asymptote at y = leading coefficient of top over leading coefficient of bottom. 1 1

Find some points on either side of each vertical asymptote xS(x)S(x) 45 Let's choose a couple of x's on the right side of the vertical asymptote We already have some points on the left side of the vertical asymptote so we can see where the function goes there 62.3

Connect points and head towards asymptotes. There should be a piece of the graph on each side of the vertical asymptote. Pass through the points and head towards asymptotes Pass through the point and head towards asymptotes Go to a function grapher or your graphing calculator and see how we did REMEMBER that x  -3 so find the point on the graph where x is -3 and make a "hole" there since it is an excluded value.

Find the equations of the horizontal asymptotes of:

Vertical Asymptotes: Horizontal Asymptotes: Holes: Intercepts:

Vertical Asymptotes: Horizontal Asymptotes: Holes: Intercepts:

Vertical Asymptotes: Horizontal Asymptotes: Holes: Intercepts:

Vertical Asymptotes: Horizontal Asymptotes: Holes: Intercepts:

Extension: The graph contains an hole at x = -3 Note: Cancelled and eliminated Extension: The graph contains an asymptote at x = 3 Note: not eliminated

Vertical Asymptotes: Horizontal Asymptotes: Holes: Intercepts:

Extension: The graph contains a hole at x = -2 Note: cancelled and eliminated

Vertical Asymptotes: Horizontal Asymptotes: Holes: Intercepts:

Graph the rational function which has the following characteristics Vert Asymp at x = 1, x = -3 Horz Asymp at y = 1 Intercepts (-2, 0), (3, 0), (0, 2) Passes through (-5, 2)

Graph the rational function which has the following characteristics Vert Asymp at x = 1, x = -1 Horz Asymp at y = 0 Intercepts (0, 0) Passes through (-0.7, 1), (0.7, -1), (-2, -0.5), (2, 0.5)