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77 Cards in this Set
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- Back
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What is the first step?
Solve: 8x+10=4x-30 |
Collect like terms:
move 4x to right side; -4x move 10 to left side; -10 |
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To clear the fraction, multiply by the LCD. What is the LCD?
(y+10)/15-1/5=(y+1)/6-1/10 |
LCD = 30
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What is the real part?
Imaginary part? Conjugate? 9-4i |
real: 9
imaginary: -4i Conjugate: +4i |
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What is the first step?
3(4x-5)-6<2x-1 |
clear the parentheses
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Solve: 8x+10=4x-30
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x = 5
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Solve:
(y+10)/15-1/5=(y+1)/6-1/10 |
y = 4
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What is the first step?
Solve: (21+9i)/(5-2i) |
multiply the numerator and the denominator by 5+2i
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What is the first step?
Second step? Solve: 2x^2=4x |
First step? divide by 2
Second step? take the square root of both sides |
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Set this up:
|2x-3| > 7 |
2x-3 > 7
2x-3 < -7 |
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What is the first step?
sqrt(1-2m^2) < 3 |
square both sides of the equation
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Does |2x-3|=-6 have a solution? If so, what is it?
If not, why? |
No, because the absolute value is always positive. It measures distance and we don't use negative numbers to measure distance.
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Solve: |3-2x|<=5
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1/2 <= x <= 4
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Solve: sqrt(1-2m)^2 <= 3
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m<=2
m<= -1 |
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Write the interval notation for
|3x-8| > 2 |
[1/2, 4]
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In the ordered pair (-4, -1),
which is the x-coordinate? which is the y-coordinate? |
x-coordinate = -4
y-coordinate = -1 |
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How do you test the equation
y=x^3 for symmetry? |
Replace x with -x
Replace y with -y Test the origin by replacing x with -x AND y with -y |
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What formula is this?
M=(x1+x2)/2 , (y1+y2)/2 |
Midpoint Formula
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What formula is this?
d = sqrt(x2-x1)^2+(y2-y1)^2 |
Distance Formula
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In the formula:
(x-h)^2 + (y-k)^2 - r^2 what do h and k represent? What does r represent? |
h and k represent the center
r is the radius |
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To find the center and radius of x^2 + y^2 + 6x - 4y = 23,
what do you do first? what do you do next? |
group terms together involving x & y
complete the square |
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Write the equation of a circle:
C = (0, 0), r = 7 |
x^2 + y^2 = 49
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Write the equation of a circle:
C = (-4, 1), r = sqrt(2) |
(x+4)^2 + (y-1)^2 = 2
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How do you find the x & y intercepts?
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x-int: Let y=0; solve for x
y int: Let x = 0; solve for y |
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Find the x & y intercept:
3x - 4y = 12 |
x-int: (4, 0)
y-int: (0, -3) |
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What is this equation called?
y = mx + b What is m? What is b? |
Slope-intercept form
m = slope b = y-intercept |
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If you were given the points:
((-3, -3) and (2, -3) to find the slope, what formula would you use? |
the slope formula:
m = (y2-y1) / (x2-x1) |
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To find the slope of
2x + 3y = 24; you do what first? What is the slope? |
Put the equation in slope-intercept form.
m = -2/3 |
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If I have a slope of -5, what is the perpendicular slope?
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perpendicular slope = 1/5
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Write the equation of the line that contains (1, 6) and (5, -2).
Write your answer in standard form and slope-intercept form. |
Standard Form: 2x + y = 8
Slope-int. Form: y = -2x + 8 |
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Given: Slope = 7/2;
y-int. = -1/3. What is the equation in standard form? |
-21x + 6y = -2
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How do you graph?
3x + 2y = 6? |
Put the equation in slope-intercept form. Graph using the slope and the y-intercept.
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What is the quadratic formula?
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x = -b+/- sqrt(b^2-4ac) / 2a
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What is the first step:
x^-4 - 8x^-2 +4 = 0 Solve for x using the quadratic formula. |
Let u = x^-2; u^2 = x^-4
2(2+/-sqrt(3)) |
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To simplify i^47, what should be done first?
Simplify: i^47 |
rewrite by removing i^2 or i^4:
i^47 = (i^2)^23 * i = (-1)*i = -i i^47 = (i^4)^11 * i^3 = (1)*(-i) = -i |
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Solve:
2.15x - 3.73(x - 0.930) = 6.11x |
x = 0.451
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What does it mean to say a function is one-to-one?
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No two ordered pairs have the same first coordinate.
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Given: (1, 2), (3, 4), (1, -5), (-3,2). Is this a function? Why?
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No
Because the ordered pairs (1, 2) & (1, -5) have the same first coordinate. |
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What is the domain and range of a function?
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Domain: the set of all x-coordinates on the graph
Range: the set of all y-coordinates on the graph. |
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Given: (1, 2), (3, 4), (1, -5), (-3,2). What is the Domain?
What is the Range? |
Domain: {(-3, 1, 3)}
Range: {(-5, 2, 4)} |
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How does the vertical line test help to determine if a graph is a function?
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When you draw a vertical line through the graph, if it touches the graph is two places, it is not a function.
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For y = f(x) + k; if k > 0 will this yield a vertical or horizontal shift?
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This will cause a vertical shift moving the graph of f(x) up 'k' units.
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For y = f(x + h); if h < 0 will this yeild a vertical or horizontal shift?
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This will cause a horizontal shift, moving the graph of f(x) to the right |h| units.
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The graph of y= - f(x) represents a reflection through which axis?
How do you obtain this? |
Reflection through the x-axis
By changing the sign of each y-coordinate in the graph of y = f(x). |
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The graph of y = f(-x) represents a reflection through which axis?
How do you obtain this? |
Reflection through the y-axis.
By changing the sign of each x-coordinate in the graph of y = f(x). |
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The graph of y = - f(-x) represents a reflection through what?
How do you obtain this? |
Reflection through the origin.
By changing the sign of the x & y coordinates of the graph of y = f(x). |
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If f(-x) = - f(x), what type of function do you have?
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An odd function
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If f(x) = f(-x), what type of function do you have?
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An even function
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The graph of x^2 is shifted three units to the right and 2 units up. What is the equation?
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f(x) = (x-3)^2 + 2
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What is the domain and range of f(x) = sqrt(x) + 3?
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Domain: [0, +oo)
Range: [3, +oo) |
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What is the domain and range of f(x) = 4 - x^2?
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Domain: (-oo, +oo)
Range: [4, -oo) |
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Given: y = A f(x); What does it mean for a graph to stretch or shrink vertically?
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If 'A' >1, you will have a vertical stretch. If 'A' is between 0 & 1, the graph will shrink. You multiply each y-coordinate by A.
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Given: y = f(Ax); What does it mean for the graph to stretch or shrink horizontally?
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If 'A' >1, you will have a horizontal shrink. If 'A' is between 0 & 1, the graph will stretch. You multiply each x-coordinate by 1/A.
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Is the function even, odd or neither? g(x) = x^4 + 3x^2
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This function is even.
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Is the function even, odd, or neither? h(x) = 3x - 4
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This function is neither.
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What is the equation? The graph is f(x) = sqrt(x) and is horizontally stretched by a factor of 0.5 and shifted two units to the left.
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f(x) = sqrt(0.5x - 2)
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True or False:
The domain of a quadratic function is the set of all real numbers. |
TRUE
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To graph f(x) = 2(x-3)^2 + 4, what is the sequence of transformations you should use?
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First, start with the graph of
f(x) = x^2. Multiplying by 2 will give a vertical stretch. Then, subtracting 3 means to move the graph 3 units to the right. Next, adding 4 outside the square means to move the graph 4 units up. |
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f(x) = a(x - h)^2 + k
What is this formula called? How is it usually found? |
This is called the vertex form; the vertex is the point (h, k).
You usually find it by completing the square. |
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x = - b/a; What is this formula used for?
What is the difference between this formula and f(x) = a(x - h)^2 + k? |
This formula is used to find the vertex of a parabola.
This formula only gives you the x-coordinate of the vertex. You will need to plug the x-value into the quadratic equation to find the y-coordinate. |
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Find the equation of the parabola with vertex (-2, 3) and x-intercept of 2.
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Using f(x) = a(x-h)^2 + k, we have: a(x + 2)^2 + 3. Since x=2 is an intercept, then f(2) shows us that a = 19. The equation of the parabola is
f(x) = 19(x + 2)^2 + 3 |
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Solve and write the answer in interval notation:
x^2 + x < 12 |
-4 < x < 3 OR
(-4, 3) |
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Solve and write the answer in interval notation:
x^2 + 3x - 10 > 0 |
(-oo, -5) U (2, +oo)
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Given: f(x) = 1/(x+2) and
g(x) = (x-5) / x. Find f/g and give the domain. |
f/g = x / (x+2)(x-5) The domain is all real numbers 'x' except
-2, 5, & 0. |
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Given: h(x) = 11 + x^2 and
k(x) = 4x - 1. Find (h o k)(x) |
16x^2 - 8x + 12
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f(x) = 2-x; g(x) = sqrt(3-x)
Find: (f+g)(3) and (fg)(-1) |
(f + g)(3) = 6
(fg)(-1) = 6 |
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If f is a one-to-one function, the the inverse of f is writtin how? Also, what does this mean?
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It is written as f^-1. It means the function is formed by reversing all the ordered pairs in f. So, (x,y) becomes (y, x)
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Are these two functions inverses of each other?
f(x) = 2/5 * (11-x) g(x) = - 5x/2 + 11 |
f and g are inverses because
f(g(x)) = x and g(f(x)) = x |
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How do you solve a system of two equations by graphing?
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Graph each equation. The point at which the graphs intersect will be the solutin of both equations.
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Solve graphically:
y + 2x = 3 y + 2x = -4 |
No solution. The graphs are parallel lines. Therefore, the system is inconsistent.
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What does it mean if you solve a system of equations and the two lines coincide?
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This means that the system has Infinitely Many Solutions. It is consistent and dependent.
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Using the substitution method, what would the 1st step look like for the system:
x + y = 6 y = x + 2 |
x + (x + 2) = 6
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Solve using substitution:
x + y = 6 y = x + 2 |
(2, 4)
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What is the best method to solve this system? Why?
2x - 3y = 0 -4x + 3y = -1 |
The Elimination Method is best. Looking at the system, if you add them, the y terms will cancel immediately.
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Solve using any method:
2x - 3y = 0 -4x + 3y = -1 |
(1/2, 1/3)
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What is the biggest mistake made in solving systems of equations?
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Once you find the answer to one variable, most forget to find the answer for the 2nd variable.
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Solve using the elimination method:
2x + 3y = 17 5x + 7y = 29 |
(-32, 27)
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Solve:
x + y + z = 4 x - 2y - z = 1 2x - y - 2z = -1 |
(2, -1, 3)
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