11th Grade Mathematics: Functions Study Guide for the SBAC

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Building and Manipulating Functions

Functions can be combined in different ways to make a single function that is equivalent (has the same input-output operation as the original functions.)

Building

Functions can be built by observing the outputs and deducing what operation is being performed on the inputs. Sometimes, more than one operation is done to produce an output. You can picture this as the output of one function being fed in as the input of a second function, producing a second output. A linear function can produce an output that is then input into an exponential function, or vice versa. Examples follow.

Combining Functions

New functions can be created by combining other functions.

For a simple linear example, suppose you can throw a ball according to the function S(t) = 40t, where S is distance and t is time.

A train can pull a flatcar with you on it according to the function \(R(t) = 50t\), where R is distance and t is time.

If you are on the moving flatcar and throw the ball forward, the flatcar’s speed will be added to the ball’s speed, and you can write a new function, \(D(t) = S(t) + R(t)\) to represent the combined speed of the train and the ball.

We could also write \(D(t) = 40t + 50t\), or \(D(t) = 90t\). This sort of combination of functions can also be done with other types of functions such as quadratic, exponential, or absolute value.

Create A Function Based On A Description

You also should be able to create a function based on a description of a relationship or a graph. For example, if a rectangle has a length that is three times its width, express the area as a function of its length, l.

\[A=lw\]

Since \(l=3w\), it follows that \(w=\frac{l}{3}\).

\[A=l\cdot\frac{l}{3}\] \[A(l)=\frac{l^{2}}{3}\]

Composition Of Functions

Another method of combining functions is called composition of functions.

Suppose you have two functions, \(f(x)=2x\) and \(g(x)=x^{3}\).

They can be combined into a new function that is written \(f(g(x))\), and read as f of g of x.

The f(x) function says to double the input and the g(x) function says to cube the input.

In our new composite function we take the output of the g(x) and use it for the input of f(x).
Start with \(f(g(x))\).

If we input x into g(x), the output is \(x^{3}\), so we then input \(x^{3}\) into f(x).

\[f(g(x))=2(g(x))\] \[f(g(x))=2(x^{3})\] \[f(g(x))=2x^{3}\]

Factors and Zeros

Factoring To Find Zeros And Vertex Of A Quadratic Function

The zeros of a function are those values of x (input values) that produce an output of zero. In other words, they tell where the graph of a function crosses the x-axis. To find the zeros, set the function equal to zero and solve for x. If the function is quadratic, you may be able to factor it to find the zeros.

So: \(x^{2}+x-6=0\)

And: \((x-2)(x+3)=0\)

\(x-2=0\) and \(x+3=0\)

\(x=2\) and \(x=-3\)

The vertex is halfway between the zeros, so in this case it would be at \(x=-0.5\), which also gives the axis of symmetry: \(x=-0.5\).

Completing The Square To Find The Vertex Of A Quadratic Function

Completing the square is another way to find the vertex. Remember that if the quadratic is in the form \(f(x)=a(x-h)^{2}+k\), the vertex of the parabola will be (h,k).

\[f(x)=x^{2}+6x+7\]

Make a space.

\[f(x)=x^{2}+6x \ \ \_\_\_\_\_+7\]

Square half of 6 and put it in the space. Then subtract 9 to stay balanced.

\[f(x)=x^{2}+6x+\underline{\;\;\;9\;\;\;\; }+7-9\] \[f(x)=(x^{2}+6x+9)-2\] \[f(x)=(x+3)^{2}-2\] \[f(x)=(x-(-3))^{2}-2\]

The vertex is (-3,-2).

Interpreting A Quadratic Function From A Description

Given a description, know how to form and interpret a quadratic function. Here is an example:
The area of a rectangle is \(88\ m^{2}\) . Find the dimensions of the rectangle if we know that one side is \(5 m\) less than two times the other side.

Assign the short side a value of \(x\). That would make the longer side \(2x-5\).
Area is a function of the lengths of the sides, so we can write it as a function of x, the short side.

\[A(x)=x(2x-5)\] \[88=2x^{2}-5x\] \[2x^{2}-5x-88=0\] \[(2x+11)(x-8)=0\] \[2x+11=0 \text{ and } x-8=0\] \[x=-\frac{11}{2} \text{ and } x=8\]

A side can’t be negative, so reject \(x=-\frac{11}{2}\).
The sides are \(x=8\) and \(2(8)-5=11\).

Completing the Square to Find Maxima or Minima

If a ball is thrown upward at 32 ft/s, its height is a function of time: \(H(t)=-16t^{2}+32t\).
What is the maximum height the ball will reach?

First, notice that the leading negative sign tells us that this parabola opens downward, and therefore, the vertex will be the maximum value of the function. Complete the square to find the vertex, (h,k).

\[H(t)=-16t^{2}+32t\] \[H(t)=-16(t^{2}-2t + \_\_\_\_)\] \[H(t)=-16(t^{2}-2t + \underline{\;\;1\;\;\;}) +16\]

Square half of -2 and put it in the blank.

Then add 16 to stay balanced.

\[H(t)=-16(t-1)^{2} +16\]

The function is now in the form of \(f(x)=a(x-h)^{2}+k\), and we can see that h=1 and k=16.

At time 1, the ball reaches a maximum height of 16 ft.

Properties of Exponents

Exponential functions can often be simplified by using the properties of exponents. Here’s an example: If you jump out of an airplane, you will pick up speed as you head for the ground, but you will quickly reach what is called terminal velocity and will then stay at that speed. This is modeled by this exponential function:

\[V(t)= 180-\frac{180}{e^{0.2t}}\]

Using the law for division of exponents, this can be transformed into a simpler form:

\[V(t)= 180-180{e^{-0.2t}}\] \[V(t)=180(1-e^{-0.2t})\]

Comparing Functions

It’s good to remember that a linear function increases or decreases by a constant difference over a constant interval (has a constant slope), whereas an exponential function increases or decreases by a constant factor over a constant interval. Also, although linear, quadratic, and exponential functions can all represent an increasing situation, an exponential function will eventually grow so fast that it will exceed the other two.

Know how to solve exponential equations using logarithms. For example, solve

\[2^{x+4}=310\]

Take the log of both sides.

\[log(2^{x+4})=log(310)\]

Rewrite using the appropriate logarithm law.

\[(x+4)(log(2))=log(310)\hspace{0.2cm}\]

Divide both sides by \(log(2)\)

\[x+4=\frac{log(310)}{log(2)}\hspace{0.2cm}\]

Subtract 4 from both sides.

\[x=\frac{log(310)}{log(2)}-4\hspace{0.2cm}\]

Use calculator to find logs.

\[x=8.276-4\hspace{0.2cm}\] \[x=4.276\hspace{0.2cm}\]

Trigonometric Functions

Trigonometric functions are used in engineering, electronics, chemistry, physics, and other technical pursuits. Check out the three skills below for a review of some of the manipulations of trig functions you should know.

6-trigonometric-function-chart.jpeg

Retrieved from: https://cnx.org/contents/_VPq4foj@11.10:F6bX9ckM@17/5-2-Unit-Circle-Sine-and-Cosine-Functions

Extend the Domain

The unit circle above shows x and y coordinates for commonly used angles from \(0^\circ\) to \(360^\circ\) \((0\text{ to } 2\pi\text{ radians })\) Using the x and y along with r=1 can give you values for any of the six trig functions in the domain \(0 \;\text{to} \;2\pi\). This domain can be extended positively or negatively as far as you like. For example, if you want \(\sin(3\pi)\) start at \(2\pi\) and go counter-clockwise an additional \(\pi\) unit and you will be at \(3\pi\) (labeled \(\pi\)). The circle shows you that the sine of \(3\pi\) is 0, the same as the sine of \(\pi\).

Likewise, if you want the tangent of \(-420^\circ\), start at \(0^\circ\), go clockwise \(-360^\circ\) and continue another \(-60^\circ\), which will put you at \(-420^\circ\), (labeled \(300^\circ\)). The tangent of \(-420^\circ\) is \(\sqrt{3}\), the same as the tangent of \(-60^\circ\) or \(300^\circ\).

7-extend-the-domain-graph.jpeg

Model Periodic Phenomena

Many phenomena are periodic and can be modeled by sinusoidal functions (or cosine functions.) Ocean tides are an example of something that can be approximately modeled this way. The graph above assumes that low tide is a height of zero and the period between high tides is 12 hours. Since the maximum is at t=0, it looks like a cosine graph, so we will use the cosine function. To write this as a cosine function, remember that the general form of the cosine function is \(f(x)=A \cos(Bx-C)+D\), where:

A is the amplitude,

the period is \(\frac{2\pi}{B}\),

\(\frac{C}{B}\) is the phase shift, and

D is the vertical shift.

By looking at the graph, we can see that:

A = 3 (from the midline to the top)

B = \(\frac{\pi}{6}\) (Use period, 12= \(\frac{2\pi}{B}\) and solve for B. )

C= 0 (No phase shift)

D= 3 (The graph has been shifted upward by 3.)

That allows us to write this function for tide movement: \(f(t)=3\cos(\frac{\pi}{6}t)+3\)

Trigonometric Identities

There are quite a few trig identities, but knowing the six reciprocal identities, two quotient identities, and three Pythagorean identities will get you pretty far with proving or evaluating identities. Useful tips for proving identities:

Start on the most complicated side.
If nothing else jumps out at you, put everything in terms of sines and cosines.
Persevere.

Here’s an example as a reminder of how to do these.

\[\begin{array}{ll} \tan^{2}x = & csc^{2}x \cdot tan^{2}x-1\\ \text{ } & \dfrac{1}{sin^{2}x} \cdot \dfrac{sin^{2}x}{cos^{2}x}-1\\ \text{ } & \dfrac{1}{cos^{2}x}-1\\ \text{ } & sec^{2}x-1\\ \tan^{2}x = & tan^{2}x\\ \end{array}\]

Other Math Abilities Tested

In addition to studying the material above, be sure to check out the other skills tested here at the end of the last page of our SBAC High School Mathematics: Numbers and Operations Study Guide. You should be fluent in all of the concepts listed there, as well.

Tackling Differently Formatted Test Items

There is important information about differently formatted test items on the SBAC exam. Go to our home page for the SBAC to read it as you prepare. Scroll down to “Tips and Tricks.”

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