Pre-Calc midterm

Being able to do the math is great but if you can't understand the question then how can you do it?

How do you algebraically determine if a function is even, odd, or neither?

Replace x with (-x), then redistribute the symbols.

What is the composition rule for inverse functions?

f(x) of g(x) should equal to x and vice versa; if they do not both equal x, then they are not inverses.

What is the first step to determining Zeros, Horizontal Asymptotes, Vertical Asymptotes, and Holes?

Factor the equation.

What is a hole in a rational function?

When the numerator and denominator have matching factors.

What is a zero in a function?

Whatever is remaining on the top of the numerator.

What constitutes a vertical asymptote?

Whatever is remaining on the bottom of the denominator.

What determines horizontal asymptotes?

Horizontal asymptotes are dependent on the leading degrees of both the numerator and denominator.

What does A > B indicate for horizontal asymptotes?

None.

What does A < B indicate for horizontal asymptotes?

y = 0.

What does A = B indicate for horizontal asymptotes?

HA present at y = C/D (the lead coefficients of the denominator and numerator).

What is the notation for the inverse function of f(x)?

f-1(x).

How do you algebraically determine the equation of an inverse function?

Replace the X value as y, then equal the equation to x and solve for y.

In the transformation ‘-2f-3(x+4))+5’, what does the first negative sign represent?

Vertical reflection over the X-axis.

In the transformation ‘-2f-3(x+4))+5’, what does the value 2 represent?

Vertical dilation over the Y-axis, multiply y values by 2.

In the transformation ‘-2f-3(x+4))+5’, what does the second negative sign represent?

Horizontal reflection over the y-axis.

In the transformation ‘-2f-3(x+4))+5’, what does the value 3 represent?

Horizontal dilation over the x-axis, divide x values by 3.

In the transformation ‘-2f-3(x+4))+5’, what does +4 represent?

Horizontal translation, 4 units to the left.

In the transformation ‘-2f-3(x+4))+5’, what does +5 represent?

Vertical translation, 5 units up.

What is the first step to synthetic division?

Set the zero of the function to not equal 0.

What is the remainder theorem?

The remainder theorem is when we take a zero of the function and substitute all X values with the said zero; it's also the same as the factor theorem.

How can we confirm our remainder theorem?

With synthetic division.

What must we remember when writing a polynomial function of minimum degree with given zeros?

Our zeros are (x - 2)(x - (3 + i))(x - (3 - i)).

How do you determine the number of complex zeros a polynomial function has?

The highest degree of the polynomial indicates how many complex zeros are present.

What is P/Q used for?

To determine the list of potential real rational zeros of the polynomial.

What do even and odd multiplicities do on a graph?

An even multiplicity will bounce off the x-axis, while an odd multiplicity will cross through the x-axis.

What can be used to find all the complex zeros?

Quadratic formula and factor by grouping.

How do you simplify complex expressions?

Multiply like values and stop there.

What is the formula for the exponential function?

f(x) = AB^x.

How do you determine the formula of an exponential function using a table?

A is the value of f(x) when x is 0; B is A(f(x)) where x is 1.

Describe the identity function.

A diagonal line through the origin.

Describe the squaring function.

A parabola (U-shape) that is symmetric around the Y-axis.

Describe the cubing function.

An S-shaped curve that passes through the origin.

Describe the absolute value function.

A ‘V’ shape.

Describe the Exponential function.

Rapid growth.

Describe the reciprocal function.

A hyperbola with vertical and horizontal asymptotes.

Describe the square root function.

An increasing curve starting at the origin.

Describe the sine function.

A squiggly line that crosses through the origin.

Describe the cosine function.

A squiggly line that is symmetric along the y-axis.

Describe the logistic function.

A very stretched out S-shaped curve.

Describe the natural function.

A slow, increasing curve with a vertical asymptote at x=0.

Describe the greatest integer function.

A staircase.

What function is represented by f(x) = x?

Identity function.

What function is represented by f(x) = |x|?

Absolute value function.

What function is represented by f(x) = √x?

Square root function.

What function is represented by f(x) = x²?

Squaring function.

What function is represented by f(x) = 1/x?

Reciprocal function.

What function is represented by f(x) = sin(x)?

Sine function.

What function is represented by f(x) = e^x?

Exponential function.

What function is represented by f(x) = 1 / (1 + e^(-x))?

Logistic function.

What function is represented by f(x) = x³?

Cubing function.

What function is represented by f(x) = cos(x)?

Cosine function.

What function is represented by f(x) = int(x)?

Greatest integer function.

What function is represented by f(x) = ln(x)?

Natural log function.

Is the identity function bounded?

Unbounded.

Is the squaring function bounded?

Bounded below.

Is the cubing function bounded?

Unbounded.

Is the absolute value function bounded?

Bounded below.

Is the exponential function bounded?

Bounded below.

Is the reciprocal function bounded?

Unbounded.

Is the square root function bounded?

Bounded below.

Is the Sine function bounded?

Bounded.

Is the cosine function bounded?

Bounded.

Is the logistic function bounded?

Bounded.

Is the natural log function bounded?

Bounded above.

Is the greatest integer function bounded?

Unbounded.

What is the equation for the exponential function?

f(x) = AB^x.

What is the first step to finding an exponential function?

Find the value of A when x is 0.

What is the second step to finding an exponential function?

The equation should equal whatever value when x is 1.

What is the compound interest formula?

P(1 + R/n)^(nt).

What happens to the value of n as you compound more?

The larger n gets, the more you compound.

What does a > 0 and k > 0 indicate?

Growth.

What does a > 0 and k < 0 indicate?

Decay.

How do you solve logistic functions?

Use the CAB method.

What is the CAB equation?

C / (1 + AB^x).

What does the equivalent of y = b^x translate to in logs?

y - 1 = log_b(x).

How to find inverse functions from log?

Arrow from b to 0 and arrow from 0 to 1.

What is log_10(100)?

2 (base 10 is ignorable).

What does ln(e√e) = x equal to?

x = √e.

What does e^(ln(4)) = z equal to?

ln(z) = ln(4), so z = 4.

How do you find the domain for a square root equation?

set the equation to greater than or equal to zero

How do you find the domain for a reciprocal equation?

set the equation so it can't be equal to zero