### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Graph A Function

The Y axis is a _______________ of the function Y = 1/X

**Possible Answers:**

Zero solution

Asymptote

Equation

Vertical slope

**Correct answer:**

Asymptote

A line is an asymptote in a graph if the graph of the function nears the line as X or Y gets larger in absolute value.

### Example Question #12 : Graphing

A sociologist studying communication wants to better understand groups that associate with outsiders. He constructs graphs with tables and is looking for populations where every input has one independent output, which means that people are associating with strangers. Which graph illustrates a population that does not associate with outsiders?

**Possible Answers:**

None of these

**Correct answer:**

This question relies on the vertical line test and the definition of a function. The sociologist wants to find populations that engage with outsiders or where an input has an independent output. This is the definition of a function; therefore, we need to use the vertical line test to determine which of the graphs is not a function (i.e. the graph that has more than one output for a given input). The vertical line test states that a graph represents a function when a vertical line can be drawn at any point in the graph and only intersect it at one point; thus, if a vertical line is drawn in a graph and it intersects it at more than one point, then the graph is not a function. The circle is the only answer choice that fails the vertical line test, and so it is not a function.

### Example Question #13 : Graphing

Which of the given functions is depicted below?

**Possible Answers:**

**Correct answer:**

The graph has x-intercepts at x = 0 and x = 8. This indicates that 0 and 8 are roots of the function.

The function must take the form y = x(x - 8) in order for these roots to be true.

The parabola opens downward, indicating a negative leading coefficient. Expand the equation to get our answer.

y = -x(x - 8)

y = -x^{2} + 8x

y = 8x - x^{2}

Therefore, the answer must be y = 8x - x^{2}

### Example Question #14 : Graphing

What is the domain of the following function:

**Possible Answers:**

x ≠ –1

x ≠ 5

x ≠ 2

x ≠ –2 and x ≠ –3

x = all real numbers

**Correct answer:**

x ≠ –2 and x ≠ –3

The denominator cannot be zero, otherwise the function is indefinite. Therefore *x* cannot be –2 or –3.

### Example Question #2 : How To Graph A Function

The figure above shows the graph of y = f(x). Which of the following is the graph of y = |f(x)|?

**Possible Answers:**

**Correct answer:**

One of the properties of taking an absolute value of a function is that the values are all made positive. The values themselves do not change; only their signs do. In this graph, none of the y-values are negative, so none of them would change. Thus the two graphs should be identical.

### Example Question #2 : How To Graph A Function

Below is the graph of the function :

Which of the following could be the equation for ?

**Possible Answers:**

**Correct answer:**

First, because the graph consists of pieces that are straight lines, the function must include an absolute value, whose functions usually have a distinctive "V" shape. Thus, we can eliminate f(x) = x^{2} – 4x + 3 from our choices. Furthermore, functions with x^{2} terms are curved parabolas, and do not have straight line segments. This means that f(x) = |x^{2} – 4x| – 3 is not the correct choice.

Next, let's examine f(x) = |2x – 6|. Because this function consists of an abolute value by itself, its graph will not have any negative values. An absolute value by itself will only yield non-negative numbers. Therefore, because the graph dips below the x-axis (which means f(x) has negative values), f(x) = |2x – 6| cannot be the correct answer.

Next, we can analyze f(x) = |x – 1| – 2. Let's allow x to equal 1 and see what value we would obtain from f(1).

f(1) = | 1 – 1 | – 2 = 0 – 2 = –2

However, the graph above shows that f(1) = –4. As a result, f(x) = |x – 1| – 2 cannot be the correct equation for the function.

By process of elimination, the answer must be f(x) = |2x – 2| – 4. We can verify this by plugging in several values of x into this equation. For example f(1) = |2 – 2| – 4 = –4, which corresponds to the point (1, –4) on the graph above. Likewise, if we plug 3 or –1 into the equation f(x) = |2x – 2| – 4, we obtain zero, meaning that the graph should cross the x-axis at 3 and –1. According to the graph above, this is exactly what happens.

The answer is f(x) = |2x – 2| – 4.

### Example Question #1 : How To Graph A Function

Which of the following could be a value of for ?

**Possible Answers:**

**Correct answer:**

The graph is a down-opening parabola with a maximum of . Therefore, there are no y values greater than this for this function.

### Example Question #1 : How To Graph A Function

What is the equation for the line pictured above?

**Possible Answers:**

**Correct answer:**

A line has the equation

where is the intercept and is the slope.

The intercept can be found by noting the point where the line and the y-axis cross, in this case, at so .

The slope can be found by selecting two points, for example, the y-intercept and the next point over that crosses an even point, for example, .

Now applying the slope formula,

which yields .

Therefore the equation of the line becomes:

### Example Question #2 : Graphing

Which of the following graphs represents the y-intercept of this function?

**Possible Answers:**

**Correct answer:**

Graphically, the y-intercept is the point at which the graph touches the y-axis. Algebraically, it is the value of when .

Here, we are given the function . In order to calculate the y-intercept, set equal to zero and solve for .

So the y-intercept is at .

### Example Question #213 : Algebra

Which of the following graphs represents the x-intercept of this function?

**Possible Answers:**

**Correct answer:**

Graphically, the x-intercept is the point at which the graph touches the x-axis. Algebraically, it is the value of for which .

Here, we are given the function . In order to calculate the x-intercept, set equal to zero and solve for .

So the x-intercept is at .

### All ACT Math Resources

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