Algebra 2 Final

How would you explain simplifying a radical to a peer? Be sure to include values and variables.

3 square root 8 Find the largest perfect square that is a factor of the radicand 4 is the largest perfect square that is a factor of 8 Rewrite the radical as a product of the square root of 4 and its matching factor 2 3 square root 4 square root 2 simplify 3 times 2 square root 2 Multiply original coefficient (3) by the 'number that got out of the square root ' (2) 6 square root 2 is the answer

Define a logarithm in your own words and describe when you should use one.

A logarithm is a power to which a number must be raised in order to get some other number to show percent change or multiplicative factors.

How can you tell whether a hyperbola is horizontal or vertical?

A horizontal hyperbola has its transverse axis at y = v and its conjugate axis at x = h; a vertical hyperbola has its transverse axis at x = h and its conjugate axis at y = v.

Give an example of 2 matrices that can be multiplied, an example of 2 that can't be multiplied, and explain why.

A matrix can be multiplied by any other matrix that has the same number of rows as the first has columns. I.E. A matrix with 2 columns can be multiplied by any matrix with 2 rows. Since the number of columns in Matrix A does not equal the number of rows in Matrix B. The multiplication of A and B is undefined.

Why do absolute value equations have two answers and how do you find them both?

Because two numbers have the same absolute value

Explain the difference between a double root and a triple root and how to identify them.

Double root for Ex would be 6x6 a triple root would be 6x6x6 so the double root would be 36 and the triple root would be 216

How do you know when you need to factor an equation to solve it? (Assume there are no fractions)

Check b2−4ac: If this is negative, there are no real roots

How do conjugates help us simplify radical expressions?

Complex conjugates are helpful when one needs to simplify expressions such as (3+4i)(−5+6i) ( 3 + 4 i ) ( − 5 + 6 i ) . This is because, when we multiply the numerator and denominator of such an expression by the complex conjugate of the denominator, we get a single complex number.

What is the difference between the vertices and co-vertices on an ellipse? Be sure to mention co-vertices and vertices in your answer.

Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is a co-vertex of the ellipse.

What is an extraneous solution and why do they occur in radical equations?

Extraneous solutions are values that we get when solving equations that aren't really solutions to the equation. Extraneous Solutions occur because squaring both sides of a square root equation results in 2 solutions

Provide three tips/strategies for approaching and or working rational expressions.

Factor any factorable polynomial expressions in the numerators and the denominators. Cancel any identical factors that appear in both the numerators and the denominators of the expressions. Multiply the remaining numerators and multiply the remaining denominators.

Explain the benefits of factoring.

Factoring is an important process that helps us understand more about our equations. Through factoring, we rewrite our polynomials in a simpler form, and when we apply the principles of factoring to equations, we yield a lot of useful information. For example from the factored form, you can easily identify solutions.

Provide three tips for factoring and explain each in detail.

Find the square root, factor into 2 binomials (one plus, one minus),

Explain how the degree and leading coefficient affect the graph of a polynomial. Be sure to include the difference between odd degree and even degree functions.

If the degree is odd and the lead coefficient is positive, then the right end of the graph will point up and the left end of the graph will point down. If the degree is odd and the lead coefficient is negative, then the right end of the graph will point down and the left end of the graph will point up.

Explain what an inverse function is and how to find the equation of an inverse.

In mathematics, an inverse function is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x

Give an example of two matrices that cannot be added and explain why.

Matrices A and B cannot be added, because B has more columns than A. Matrices may be added or subtracted only if they have the same number of rows and the same number of columns.

What is an inverse matrix and what is it used for?

One of the major uses of inverses is to solve a system of linear equations. You can write a system in matrix form as AX = B.

Compare and contrast the three ways to solve a system of equations then state which you prefer. (substitution, elimination, graphing)

Set the equations equal to y to make graphing easier. ... Identify the slope and y-intercept in each equation. ... Graph the lines. substitution" works by solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation, In the elimination method, you either add or subtract the equations to get an equation in one variable. I prefer the graphing method

Explain the general process of polynomial synthetic division.

Synthetic division is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1. ... We then multiply it by the "divisor" and add, repeating this process column by column until there are no entries left.

How do you write the dimensions of a matrix? Give at least one example in your explanation.

The dimensions of a matrix are the number of rows by the number of columns. If a matrix has a rows and b columns, it is an a×b matrix. For example, the first matrix shown below is a 2×2 matrix; the second one is a 1×4 matrix; and the third one is a 3×3 matrix.

Explain what i and i2 are.

The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x2 + 1 = 0. i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. i2 = −1

Explain how to multiply matrices.

The number of columns of the 1st matrix must equal the number of rows of the 2nd matrix. And the result will have the same number of rows as the 1st matrix, and the same number of columns as the 2nd matrix.

How should someone find the product of a binomial and a trinomial?

To find the product of a binomial you multiply them together to find trinomials you would Identify a, b, and c in the trinomial ax2+bx+c. Write down all factor pairs of c. Identify which factor pair from the previous step sum up to b. Substitute factor pairs into two binomials.

When graphing quadratic functions, there were three components we looked for: the vertex, the y-intercept, and x-intercepts. Describe how to find each.

To find the vertex of a parabola, you first need to find x (or y, if your parabola is sideways) through the formula for the axis of symmetry. Then, you'll use that value to solve for y (or x if your parabola opens to the side) by using the quadratic equation. Those two coordinates are your parabola's vertex. To determine the x-intercept, we set y equal to zero and solve for x. Similarly, to determine the y-intercept, we set x equal to zero and solve for y.

Explain how to find each part of a rational graph: VA, HA, Holes, x-intercept, and y-intercept

To find the vertical asymptote(s) of a rational function, simply set the denominator equal to 0 and solve for x H.A. is y = 0 factor the numerator and denominator. If there is the same factor in the numerator and denominator, there is a hole. Set this factor equal to zero and solve. The solution is the x-value of the hole. To determine the x-intercept, we set y equal to zero and solve for x. Similarly, to determine the y-intercept, we set x equal to zero and solve for y.

Provide three tips for solving exponential and logarithmic equations.

Use Known Log Rules Express both sides in terms of the same base. Isolate the exponential and then apply the logarithm to both sides.

Explain how fractional exponents work.

When you have a fractional exponent, the numerator is the power and the denominator is the root.

How can you tell whether an ellipse is horizontal or vertical? Give an example of an equation of each.

Whichever denominator is larger determines which variable is a. If the larger number is under the x, then the ellipse is horizontal. If it is under the y then it is vertical. x+3/9 + y-5/3 would be horizontal x-1/2 + 4-3/6 would be vertical

What is the difference between an equation and an inequality?

While inequalities are used to represent the unequal relationship between a set of variables, equations are used to symbolically represent the equality of the two sets of variables used.

What is the difference between a negative exponent and a fractional exponent?

in a fractional exponent, you take 125 4/3 "take 125 to the fourth power and take the cube root of the result" in a negative exponent tells us how many times to divide a base number.

How does synthetic division help us find all the roots of a polynomial?

it helps simplify the problem so you can factor

Given the equation for a circle, (x-h)2+(y-k)2=r2, please define h, k, and r.

r is the radius, h and k are the points of the center.

Explain the solution to a system of equations in both the context of an equation. Be sure to include single solutions, infinite solutions, and no solution.

take 3x+4=8 times by -4 to get -12x-4y=-32 then add 2x+4y=12 to get -10x=-20 so x=2 If we end up with the same term on both sides of the equal sign, such as 4 = 4 or 4x = 4x, then we have infinite solutions. If we end up with different numbers on either side of the equal sign, as in 4 = 5, then we have no solutions.

Define the terms: polynomial, degree, and leading coefficient.

the degree of a polynomial is the highest of the degrees of the polynomial's monomials with non-zero coefficients. Leading coefficients are the numbers written in front of the variable with the largest exponent. ... For example, in the equation -7x^4 + 2x^3 - 11, the highest exponent is 4. The coefficient for that term is -7, which means that -7 is the leading coefficient.

Explain how you would graph this equation: f(x) = 2(x - 3)2 - 8

the positive 2 makes it go up an infinite amount, the -8 is the base then the intersecting points are (1,0)(5,0)

Explain what a complex solution is and why they are referred to as "imaginary".

they always occur in conjugate pairs. Imaginary or complex roots will occur when the value under the radical portion of the quadratic formula is negative imaginary numbers are all multiples of something called the imaginary unit, which we write with the letter i.

If both investments are made with the same rate and time, will it take longer for an investment of $1000 or $100,000 to triple? Be prepared to explain your answer or provide evidence.

they would take the same amount of time because they have the same rate and time

We learned 3 main ways to solve quadratics: factoring, quadratic formula, and using square roots. Briefly describe when each method should be used and explain which one you think is most efficient.

use square roots if the equation looks like (x+-#)=# use factoring if the equation looks factorable use quadratic formula as a last resort I think factoring is fastest because it's already set up.

What is the value of e and what is it used for?

used as the base for a logarithm it is equal to 2.718

How do square roots help us solve equations?

what we do to one side of an equation we must do to the other side as well. Since squaring a quantity and taking a square root are 'opposite' operations, we will square both sides in order to remove the radical sign and solve for the variable inside.

If you're given the co-vertices and foci, how should you find the vertices?

you count

How can you find the equation of a circle given a center and a point on the circle?

you plug in the center and the point in the equation r=(x-h)(y-k) you then add x-h and y-k together then you square those to answers then you add them together then you simplify that number which is the your radius