All SAT II Math I Resources
Example Question #1 : Graphing Quadratic Functions
Give the -coordinate of the vertex of the parabola of the function
The -coordinate of the vertex of a parabola of the form
The -coordinate is therefore :
, which is the correct choice.
Example Question #2 : Graphing Quadratic Functions
Give the -intercept(s) of the parabola of the equation
The parabola has no -intercept.
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 :
The parabola has as its two intercepts the points and .
Example Question #3 : Graphing Quadratic Functions
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.
Example Question #4 : Graphing Quadratic Functions
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 .
Example Question #5 : Graphing Quadratic Functions
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.
Example Question #6 : Graphing Quadratic Functions
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
Example Question #1 : Graphing Polynomials
Which is the graph of ?
Think of the graph of :
Constants within the parentheses will shift the parabola to the left and right, while terms outside of the parentheses will shift it vertically.
In our equation, there is a -2 term outside the parentheses. This will shift the graph down by 2 units.
The graph of will look like this:
There is also a constant within the parentheses, –1. This will shift the graph to the right by 1 unit.
Therefore will generate a graph like this:
Example Question #7 : Graphing Quadratic Functions
None of them
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 .
Example Question #8 : Graphing Quadratic Functions
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 .
Example Question #422 : Functions And Graphs
What is the vertex of the function ? Is it a maximum or minimum?
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.