Category Archives: MAC1105

1.1 Linear Equations, 4.1 Linear Functions and Their Properties

Solving Linear Equations: Two Step Equations

July 16, 2018 admin

Practice Problems

Addition and Multiplication Principles Combined

Solutions to Practice Problems

Addition and Multiplication Principles Combined Solutions

5.1 Exponential Functions, 6.3 Exponential Functions

Exponents: Simplify

July 16, 2018 admin

Example: Rewrite the expression in the form 5^p where p is an algebraic expression.

25/root{3}{5}

Solution:

The original expression
Rewrite the numerator with a base of 5.	$25/root{3}{5}=5^2/root{3}{5}$
Rewrite the radical as a fractional exponent. In general $root{n}{x^m}=x^{m/n}$ . For this example $root{3}{5}= 5^{1/3}$	$5^2/root{3}{5}=5^2/5^{1/3}$
Use the quotient rule for exponents to subtract the exponents. Quotient Rule for Exponents $b^m/b^n=b^{m-n}$	$5^2/5^{1/3}=5^{2-1/3}$
Simplify by getting a common denominator and combining the fractions.	$5^{2-1/3}=5^{6/3-1/3}=5^{5/3}$

3.1 Functions, 3.1 Relations and Functions

Finding Domain: Radical/Root Function (Even Root)

July 12, 2018 admin

Example: Classify the function as a polynomial function, rational function, or root function, and then find the domain. Write the domain interval notation and set builder notation.

f(x) = root{6}{2-x}

Solution:

Classify the Function

Formal Definition	Practical Way of Identifying
Polynomial Function A polynomial function is a function of the form $f(x)=a_n x^n+a_{n-1} x^{n-1}+...+a_1 x+a_0$ where n is a non-negative integer {0, 1, 2, 3, 4, …} and the coefficients $a_n , a_{n-1,...,a_1,a_0$ are from the real numbers.	Look for the variables to be in the numerator. (If there is no fraction at all, they are in the numerator.) The variable should not be inside a radical or absolute value. The powers or exponents on the variables should be whole numbers. Whole numbers come from this list {0, 1, 2, 3, 4, …}
Rational Function A rational function is a function of the form where and are polynomial functions and is not equal to zero.	The numerator and denominator are polynomials. Most functions with variables in the denominator are considered Rational Functions but there are exceptions.
Root Function (even index) A root function is a function of the form where n is an even positive integer greater than or equal to 2.	The variable is inside or underneath a radical. The index of the radical is an even number. {2, 4, 6, 8, …} The square root is an even index although the index is not written.
Root Function (odd index) A root function is a function of the form where n is an odd positive integer greater than or equal to 2.	The variable is inside or underneath a radical. The index of the radical is an odd number. {3, 5, 7, 9, …} The cube root is an odd index.

Since the function f(x) = root{6}{2-x} has a radical and the index is even. This function is a root function.

Find the Domain of a Root Function (Even Index)

Taking the even root of a negative number results in a complex or imaginary number. Since we are interested in real function values, we would like the expression inside the radical to be non-negative ( zero or positive) The root function is defined for any value of the variable where the expression under the radical is non-negative (zero or positive). Find these values by creating an inequality to solve. The inequality is the expression under the radical greater than or equal to zero.

Set the expression under the radical greater than or equal to zero.	0
Solve the inequality. This inequality is a linear inequality and can be solved by isolating the variable on one side.	0
Solve by isolating the variable. Start by subtracting 2 on both sides.
Isolate the variable. Continue by dividing both sides by -1. Be sure to reverse the inequality symbol since you are dividing both sides by a negative.

The function values where x<=2 are defined for f(x) = root{6}{2-x} .

In set builder notation, the domain is .

In interval notation, the domain is ( - infty, 2 ]

1.9 Polynomial and Rational Inequalities, 5.6 Polynomial and Rational Inequalities

Inequalities: Rational Inequalities

July 11, 2018 admin

5.5 Applications of Exponential and Logarithmic Functions, 6.7 Financial Models, Applications of Exponential Functions

Application of Exponential Functions: Doubling Time

July 11, 2018 admin

Example:

How long does it take for an investment to double if it is invested at 18% compounded continuously?

Solution:

Since this question involve continuous compound interest, we will use the associated formula.

$A=Pe^{rt}$

We are given that the interest rate is 18% or 0.18. This tells me that when r=0.18 Since we are looking for the doubling time, A will be 2 times P. I can write that in symbols A=2P.

Substitute these values into the continuous compound formula and solve for the interest rate.

Continuous compound formula	$A=Pe^{rt}$
Substitute the values of r and A into the formula	$A=Pe^{rt}$ $2P=Pe^{0.18*t}$ $2P=Pe^{0.18t}$
Solve for t by dividing both sides by P and simplifying	$2P=Pe^{0.18t}$ ${2P}/P={Pe^{0.18t}}/P$ $2=e^{0.18t}$
Solve for t by taking the log of both sides.	$2=e^{0.18t}$ $ln 2=ln e^{0.18t}$
Solve for t by using the power rule and simplifying	$ln 2=ln e^{0.18t}$
Solve for t by dividing both sides by 0.18 and simplifying
Find the value in the calculator
Write the answer rounded to two decimal places

It will take 3.85 years to double your money when interest is compounded continuously at 18%.

If you need to write this in years and months, you will need to convert the 0.85 to months. Since there are 12 months in a year, multiply 0.85 by 12 to get 10.2. I will round to the nearest months to get 10.

It will take 3 years and 10 months to double your money when interest is compounded continuously at 18%.

Here is a video that is similar except that you are looking for the investment to triple.

4.1 Quadratic Functions, 4.3 Quadratic Functions and Their Properties

Analyze a Quadratic Function in Vertex Form

July 3, 2018 admin

Example: Given the quadratic function in vertex form, state the domain, range, vertex, x-intercepts, y-intercept, the orientation (opens up or opens down), and the axis of symmetry. Finally graph the function.

$f(x)=-{1/4}(x-1)^2+5$

Solution:

Find the vertex.

When the quadratic function is written in standard form you can identify the vertex as (h,k).

f(x)=a(x-h)^2+k

$f(x)=-{1/4}(x-1)^2+5$

The vertex of the quadratic function is (1,5) .

Find the orientation.

The leading coefficient of the quadratic function is negative so the parabola opens down.

Find the y-intercept.

To find a y-intercept let x=0.

Start with the original function.	$f(x)=-{1/4}(x-1)^2+5$
To find a y-intercept let x=0.	$f(0)=-{1/4}(0-1)^2+5$
Simplify using order of operations. First complete operations inside parenthesis.	$f(0)=-{1/4}(0-1)^2+5$ $f(0)=-{1/4}(-1)^2+5$
Apply the exponent.	$f(0)=-{1/4}(-1)^2+5$
Multiply.
Get a common denominator and combine fractions

The y-intercept is (0,19/4) .

Find the x-intercept.

Start with the original function.	$f(x)=-{1/4}(x-1)^2+5$
To find an x-intercept let y=0 or f(x)=0.	$0=-{1/4}(x-1)^2+5$
Isolate the square by subtracting 5 on both sides. Then simplify.	$0=-{1/4}(x-1)^2+5$ $0-5=-{1/4}(x-1)^2+5-5$ $-5=-{1/4}(x-1)^2$
Isolate the square by multiplying by -4 on both sides. Then simplify.	$-5=-{1/4}(x-1)^2$ $-5(-4)=(-4)-{1/4}(x-1)^2$
Get rid of the square by square rooting both sides.	$sqrt{20}=sqrt{(x-1)^2}$
Simplify the radical by finding a perfect square factor of 20.
Isolate x by adding 1 to both sides. Then simplify.
Rewrite as two separate answers.	or

The x-intercepts are (1 + 2 sqrt{5},0) and (1 - 2 sqrt{5},0)

Find the axis of symmetry.

The axis of symmetry is a vertical line that passes through the vertex. Since the vertex is (1,5) the axis of symmetry is x=1 .

Graph the function.

Plot the intercepts and the vertex.

Since the vertex is the highest point we can draw the parabola using the peak at the vertex.

Find the domain and range.

The domain is (- infty, infty) .

The range is - infty, 5 {}{]} .

Here is a video example analyzing a quadratic function in vertex form.

4.2 Applications and Modeling of Quadratic Functions, 4.4 Build Quadratic Models from Verbal Descriptions and from Data

Application: Quadratic Minimimization

July 2, 2018 admin

Problem:

A 28 inch wire is to be cut. One piece is to be bent into the shape of a square, whereas the other piece is the bent into the shape of a rectangle whose length is twice the width. Find the width of the rectangle that will minimize the total area.

Solution:

First draw a picture of the shapes the wire will make and label the sides.

The square has equal sides. Label the side as an unknown quantity x.

The perimeter of the square is and the area of the square is x^2 .

The rectangle described as a length that is twice the width. Label the sides as y and 2y.

The perimeter of the rectangle is and the area of the rectangle is 2y^2 .

Since the 28 inch piece of wire will be cut and used to form the square and rectangle the total perimeter of the two shapes is 28 inches.

P=4x+6y=28

The total area is to be minimized.

A=2y^2+x^2

To minimize the area there must be only one variable in the expression. Use the perimeter equation to reduce the number of variables.

Perimeter Equation
Solve for one of the variables. In this example, solve for x by first subtracting 6y from both sides. Simplify.
Continue to solve for x by dividing both sides by 4 and simplifying.

Substitute into the area expression to reduce from having two variables to have one variable.

Area Expression
Substitute for x.	$A=2y^2+(7-{3/2}y)^2$
Simplify by applying the exponent and simplifying.	$A=2y^2+(7-{3/2}y)^2$ $A=2y^2+(7-{3/2}y)(7-{3/2}y)$ $A=2y^2+49-{21/3}y-{21/3}y+{9/4}y^2$ $A=49-{42/2}y+{17/4}y^2$ $A=49-21y+{17/4}y^2$

$A(y)={17/4}y^2-21y+49$ is in the form of a quadratic function. The graph of a quadratic function is a parabola. This quadratic function has a leading coefficient of 17/4 and since it is positive means that the parabola is opening up.

Find the vertex to find the minimum value.

The formula for the x coordinate of the vertex
For a quadratic function a is the coefficient of the square term, b is the coefficient of the linear term, and c is the constant	$A(y)={17/4}y^2-21y+49$
Substitute the values of a and b into the formula
Simplify with a calculator

Round to the nearest tenth and y=2.5 .

The total area of the square and rectangle is minimized when the width of the rectangle is 2.5 inches.

3.4 Transformation of Functions, 3.5 Graphing Techiques: Transformations

Graphing by Transformations: Quadratic

June 29, 2018 admin

Example: For the function below. Graph using transformations.

f(x)=-(x-3)^2+4

First we must examine the base function y=x^2

Graph using plotting points. We can use the standard set of x-values to find ordered pairs. Substitute the standard set of x-values into the base function to get the base graph.