How to graph a function - ACT Math
Card 0 of 12
Suppose 
.
To obtain the graph of 
, shift the graph 
a distance of 
units .
Suppose
To obtain the graph of
There are four shifts of the graph y = f(x):
y = f(x) + c shifts the graph c units upwards.
y = f(x) – c shifts the graph c units downwards.
y = f(x + c) shifts the graph c units to the left.
y = f(x – c) shifts the graph c units to the right.
There are four shifts of the graph y = f(x):
y = f(x) + c shifts the graph c units upwards.
y = f(x) – c shifts the graph c units downwards.
y = f(x + c) shifts the graph c units to the left.
y = f(x – c) shifts the graph c units to the right.
Compare your answer with the correct one above
The Y axis is a _______________ of the function Y = 1/X
The Y axis is a _______________ of the function Y = 1/X
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.
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.
Compare your answer with the correct one above
Which of the following graphs does NOT represent a function?
Which of the following graphs does NOT represent a function?
This question relies on both the vertical-line test and the definition of a function. 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 every point in the graph and only intersect it at one point; thus, if a vertical line is drawn in a graph and it intersects that graph 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.
This question relies on both the vertical-line test and the definition of a function. 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 every point in the graph and only intersect it at one point; thus, if a vertical line is drawn in a graph and it intersects that graph 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.
Compare your answer with the correct one above
Which of the given functions is depicted below?
Which of the given functions is depicted below?
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 = -x2 + 8x
y = 8x - x2
Therefore, the answer must be y = 8x - x2
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 = -x2 + 8x
y = 8x - x2
Therefore, the answer must be y = 8x - x2
Compare your answer with the correct one above
The figure above shows the graph of y = f(x). Which of the following is the graph of y = |f(x)|?
The figure above shows the graph of y = f(x). Which of the following is the graph of y = |f(x)|?
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.
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.
Compare your answer with the correct one above
Below is the graph of the function 
:
Which of the following could be the equation for 
?
Below is the graph of the function
Which of the following could be the equation for
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) = x2 – 4x + 3 from our choices. Furthermore, functions with x2 terms are curved parabolas, and do not have straight line segments. This means that f(x) = |x2 – 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.
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) = x2 – 4x + 3 from our choices. Furthermore, functions with x2 terms are curved parabolas, and do not have straight line segments. This means that f(x) = |x2 – 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.
Compare your answer with the correct one above
What is the domain of the following function:
What is the domain of the following function:
The denominator cannot be zero, otherwise the function is indefinite. Therefore x cannot be –2 or –3.
The denominator cannot be zero, otherwise the function is indefinite. Therefore x cannot be –2 or –3.
Compare your answer with the correct one above
Which of the following could be a value of 
for 
?
Which of the following could be a value of
The graph is a down-opening parabola with a maximum of 
. Therefore, there are no y values greater than this for this function.
The graph is a down-opening parabola with a maximum of
Compare your answer with the correct one above
What is the equation for the line pictured above?
What is the equation for the line pictured above?
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:
A line has the equation
The
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:
Compare your answer with the correct one above
Which of the following graphs represents the y-intercept of this function?
Which of the following graphs represents the y-intercept of this function?
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 
.
Graphically, the y-intercept is the point at which the graph touches the y-axis. Algebraically, it is the value of
Here, we are given the function
So the y-intercept is at
Compare your answer with the correct one above
Which of the following graphs represents the x-intercept of this function?
Which of the following graphs represents the x-intercept of this function?
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 
.
Graphically, the x-intercept is the point at which the graph touches the x-axis. Algebraically, it is the value of
Here, we are given the function
So the x-intercept is at
Compare your answer with the correct one above
Which of the following represents 
?
Which of the following represents
A line is defined by any two points on the line. It is frequently simplest to calculate two points by substituting zero for x and solving for y, and by substituting zero for y and solving for x.
Let 
. Then
So our first set of points (which is also the y-intercept) is 
Let 
. Then
So our second set of points (which is also the x-intercept) is 
.
A line is defined by any two points on the line. It is frequently simplest to calculate two points by substituting zero for x and solving for y, and by substituting zero for y and solving for x.
Let
So our first set of points (which is also the y-intercept) is
Let
So our second set of points (which is also the x-intercept) is
Compare your answer with the correct one above