Graphing Quadratic Functions - SAT Subject Test in Math I
Card 0 of 11
Which of the following graphs matches the function 
?
Which of the following graphs matches the function
Start by visualizing the graph associated with the function 
:
Terms within the parentheses associated with the squared x-variable will shift the parabola horizontally, while terms outside of the parentheses will shift the parabola vertically. In the provided equation, 2 is located outside of the parentheses and is subtracted from the terms located within the parentheses; therefore, the parabola in the graph will shift down by 2 units. A simplified graph of 
looks like this:
Remember that there is also a term within the parentheses. Within the parentheses, 1 is subtracted from the x-variable; thus, the parabola in the graph will shift to the right by 1 unit. As a result, the following graph matches the given function 
:
Start by visualizing the graph associated with the function
Terms within the parentheses associated with the squared x-variable will shift the parabola horizontally, while terms outside of the parentheses will shift the parabola vertically. In the provided equation, 2 is located outside of the parentheses and is subtracted from the terms located within the parentheses; therefore, the parabola in the graph will shift down by 2 units. A simplified graph of
Remember that there is also a term within the parentheses. Within the parentheses, 1 is subtracted from the x-variable; thus, the parabola in the graph will shift to the right by 1 unit. As a result, the following graph matches the given function
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Find the vertex form of the following quadratic equation:
Find the vertex form of the following quadratic equation:
Factor 2 as GCF from the first two terms giving us:
Now we complete the square by adding 4 to the expression inside the parenthesis and subtracting 8 ( because 
) resulting in the following equation:
which is equal to
Hence the vertex is located at
Factor 2 as GCF from the first two terms giving us:
Now we complete the square by adding 4 to the expression inside the parenthesis and subtracting 8 ( because
which is equal to
Hence the vertex is located at
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What is the vertex of the function 
? Is it a maximum or minimum?
What is the vertex of the function
The equation of a parabola can be written in vertex form: 
.
The point 
in this format is the vertex. If 
is a postive number the vertex is a minimum, and if 
is a negative number the vertex is a maximum.
In this example, 
. The positive value means the vertex is a minimum.
The equation of a parabola can be written in vertex form:
The point
In this example,
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Based on the figure below, which line depicts a quadratic function?
Based on the figure below, which line depicts a quadratic function?
A parabola is one example of a quadratic function, regardless of whether it points upwards or downwards.
The red line represents a quadratic function and will have a formula similar to 
.
The blue line represents a linear function and will have a formula similar to 
.
The green line represents an exponential function and will have a formula similar to 
.
The purple line represents an absolute value function and will have a formula similar to 
.
A parabola is one example of a quadratic function, regardless of whether it points upwards or downwards.
The red line represents a quadratic function and will have a formula similar to
The blue line represents a linear function and will have a formula similar to
The green line represents an exponential function and will have a formula similar to
The purple line represents an absolute value function and will have a formula similar to
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All of the following are equations of down-facing parabolas EXCEPT:
All of the following are equations of down-facing parabolas EXCEPT:
A parabola that opens downward has the general formula
,
as the negative sign in front of the 
term makes flips the parabola about the horizontal axis.
By contrast, a parabola of the form 
rotates about the vertical axis, not the horizontal axis.
Therefore, 
is not the equation for a parabola that opens downward.
A parabola that opens downward has the general formula
as the negative sign in front of the
By contrast, a parabola of the form
Therefore,
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Consider the equation:
The vertex of this parabolic function would be located at:
Consider the equation:
The vertex of this parabolic function would be located at:
For any parabola, the general equation is
, and the x-coordinate of its vertex is given by
.
For the given problem, the x-coordinate is
.
To find the y-coordinate, plug 
into the original equation:
Therefore the vertex is at 
.
For any parabola, the general equation is
For the given problem, the x-coordinate is
To find the y-coordinate, plug
Therefore the vertex is at
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In which direction does graph of the parabola described by the above equation open?
In which direction does graph of the parabola described by the above equation open?
Parabolas can either be in the form
for vertical parabolas or in the form
for horizontal parabolas. Since the equation that the problem gives us has a y-squared term, but not an x-squared term, we know this is a horizontal parabola. The rules for a horizontal parabola are as follows:
- If
, then the horizontal parabola opens to the right.
- If
, then the horizontal parabola opens to the left.
In this case, the coefficient in front of the y-squared term is going to be positive, once we isolate x. That makes this a horizontal parabola that opens to the right.
Parabolas can either be in the form
for vertical parabolas or in the form
for horizontal parabolas. Since the equation that the problem gives us has a y-squared term, but not an x-squared term, we know this is a horizontal parabola. The rules for a horizontal parabola are as follows:
- If
- If
In this case, the coefficient in front of the y-squared term is going to be positive, once we isolate x. That makes this a horizontal parabola that opens to the right.
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Give the 
-coordinate of the vertex of the parabola of the function
Give the
The 
-coordinate of the vertex of a parabola of the form
is
.
Set 
:
The 
-coordinate is therefore 
:
, which is the correct choice.
The
is
Set
The
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Give the 
-intercept(s) of the parabola of the equation
Give the
Set 
and solve for 
:
The terms have a GCF of 2, so
The trinomial in parentheses can be FOILed out by noting that 
and 
:
Set each of the linear binomials to 0 and solve for 
:
or
The parabola has as its two intercepts the points 
and 
.
Set
The terms have a GCF of 2, so
The trinomial in parentheses can be FOILed out by noting that
Set each of the linear binomials to 0 and solve for
or
The parabola has as its two intercepts the points
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Which of the following parabolas is downward facing?
Which of the following parabolas is downward facing?
We can determine if a parabola is upward or downward facing by looking at the coefficient of the 
term. It will be downward facing if and only if this coefficient is negative. Be careful about the answer choice 
. Recall that this means that the entire value inside the parentheses will be squared. And, a negative times a negative yields a positive. Thus, this is equivalent to 
. Therefore, our answer has to be 
.
We can determine if a parabola is upward or downward facing by looking at the coefficient of the
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How many points of intersection could two distinct quadratic functions have?
. 
. 
. 
How many points of intersection could two distinct quadratic functions have?
An intersection of two functions is a point they share in common. A diagram can show all the possible solutions:
Notice that:
and 
intersect 
times
and 
intersect 
time
and 
intersect 
times
The diagram shows that 
, 
, and 
are all possible.
An intersection of two functions is a point they share in common. A diagram can show all the possible solutions:
Notice that:
The diagram shows that
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