Graphical Representation of Functions - SAT Math
Card 0 of 20
Which of the following represents a quadratic equation with its zeros at 
and 
?
Which of the following represents a quadratic equation with its zeros at 
The important first step of creating a quadratic equation from its zeros is knowing what a zero really is. A zero of a quadratic (or polynomial) is an x-coordinate at which the y-coordinate is equal to 0.
We find that algebraically by factoring quadratics into the form 
, and then setting 
equal to 
and 
, because in each of those cases and entire parenthetical term would equal 0, and anything times 0 equals 0.
Here the question gives you a head start: we know that the numbers 4 and 5 can go in the 
and 
spots, because if so we'll have found our zeros. So we can set up the equation:
This satisfies the requirements of zeros, but now we need to expand this equation using FOIL to turn it into a proper quadratic. That means that our quadratic is:
And when we combine like terms it's:
The important first step of creating a quadratic equation from its zeros is knowing what a zero really is. A zero of a quadratic (or polynomial) is an x-coordinate at which the y-coordinate is equal to 0.
We find that algebraically by factoring quadratics into the form 
Here the question gives you a head start: we know that the numbers 4 and 5 can go in the 
This satisfies the requirements of zeros, but now we need to expand this equation using FOIL to turn it into a proper quadratic. That means that our quadratic is:
And when we combine like terms it's:
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How many distinct zeros does the function 
have?
How many distinct zeros does the function 
The zeros of a function are points at which the function crosses the x-axis, or perhaps more simply points at which the function is equal to 0. So to solve for those, set the function equal to zero and then solve it like you would a quadratic. Here that gives you:
You can then factor the common 
term:
And then factor the quadratic within parentheses. Note that this quadratic is one of the common perfect square quadratics:
or 
The solutions to this equation, then, are 
and 
. Note that the question asks you for how many distinct zeros the function has, so you cannot count 
twice. The answer, then, is 2.
The zeros of a function are points at which the function crosses the x-axis, or perhaps more simply points at which the function is equal to 0. So to solve for those, set the function equal to zero and then solve it like you would a quadratic. Here that gives you:
You can then factor the common 
And then factor the quadratic within parentheses. Note that this quadratic is one of the common perfect square quadratics:
The solutions to this equation, then, are 
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Which of the following equations belongs to a polynomial function with zeros at 
and 
?
Which of the following equations belongs to a polynomial function with zeros at 
A function with zeros at 2 and 7 would factor to 
, where it is important to recognize that 
is the coefficient. So while you might be looking to simply expand 
to 
, note that none of the options with a simple 
term (and not 
) directly equal that simple quadratic when set to 0.
However, if you multiply 
by 2, you get 
.
A function with zeros at 2 and 7 would factor to 
However, if you multiply 
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How many distinct zeros does the function 
have?
How many distinct zeros does the function 
When you're graphing a function, the zeros of the function are the points at which the function crosses the x-axis. What that really means is that the value of the function is zero at those points, so to solve for those zeros algebraically you can just set the function equal to zero and solve. Here that would mean:
So factor the common 
term to get:
And then factor the quadratic within parentheses:
You can then see that there are three values of 
that would make this equation true: 
and 
. The answer is therefore 3.
When you're graphing a function, the zeros of the function are the points at which the function crosses the x-axis. What that really means is that the value of the function is zero at those points, so to solve for those zeros algebraically you can just set the function equal to zero and solve. Here that would mean:
So factor the common 
And then factor the quadratic within parentheses:
You can then see that there are three values of 
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What is the sum of all unique zeros to the function 
?
What is the sum of all unique zeros to the function 
The zeros of a function are the x-values at which the function is equal to zero. So to solve for the zeros, set the function equal to zero. That would give you here:
Then you can factor the common 
to get:
And then factor like you would a quadratic:
Or, more succinctly formatted:
This means that the zeros are at 
and 
. Now, importantly, look at what the question asks for. It wants the sum (add the zeros) of all UNIQUE zeros, meaning you should not count 
twice. The sum then is 
.
The zeros of a function are the x-values at which the function is equal to zero. So to solve for the zeros, set the function equal to zero. That would give you here:
Then you can factor the common 
And then factor like you would a quadratic:
Or, more succinctly formatted:
This means that the zeros are at 
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The function 
does not have any zeros for its domain of all real numbers. Which of the following must be true?
The function 
The zeros of a function are points at which the function is equal to zero. Here you're told that 
does not equal zero for any real number values of 
and you're asked to determine what that means for 
. Note that since 
cannot be negative, 
would have to be negative in order for the 
term to reduce the other term, 
to zero. Since you know that this function never equals zero, you can conclude that 
is not negative, which means that it is greater than or equal to zero.
The zeros of a function are points at which the function is equal to zero. Here you're told that 
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What is the sum of all zeros to the function 
?
What is the sum of all zeros to the function 
The zeros of a function are the x-values at which the function itself is equal to zero, so you will generally want to solve these problems by setting the function equal to zero and then factoring like a quadratic. Here that means you would start with:
And then factor the common 
term:
You now know that one of the solutions is 
, as that would mean that the entire parenthetical term would be multiplied by zero. But you still need to work within the parentheses to factor that quadratic. You should see some helpful factoring clues: when you factor into two parentheticals, the numeric terms have to multiply to -1, meaning that you will likely have one be 1 and the other -1. And the first terms need to multiply to 
meaning that your first terms will be 
and 
. So you can set up your parentheses as:
, where one 
will have a + and one will have a - sign.If you play with the options to see what will work to give you the 
middle term in 
, you'll see that the proper factorization is:
This means that the solutions for 
are 
and 
. The sum of these solutions, then, is 
.
The zeros of a function are the x-values at which the function itself is equal to zero, so you will generally want to solve these problems by setting the function equal to zero and then factoring like a quadratic. Here that means you would start with:
And then factor the common 
You now know that one of the solutions is 
This means that the solutions for 
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How many distinct zeros does the function 
have?
How many distinct zeros does the function 
The zeros of a function are essentially points (or at least the x-values of the points) at which the function is equal to zero. So to solve for the zeros of a function, first set that function itself equal to zero. Here that would mean:
Then factor like you would a quadratic; since you have it set to zero, if any multiplicative term equals zero then the "equals zero" will hold for the whole equation. First you can factor the common 
term:
And then you can factor the quadratic within:
This then means that the zeros for this function are at 
and 
, meaning that this function has three distinct zeros.
The zeros of a function are essentially points (or at least the x-values of the points) at which the function is equal to zero. So to solve for the zeros of a function, first set that function itself equal to zero. Here that would mean:
Then factor like you would a quadratic; since you have it set to zero, if any multiplicative term equals zero then the "equals zero" will hold for the whole equation. First you can factor the common 
And then you can factor the quadratic within:
This then means that the zeros for this function are at 
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How many unique zeros does the function 
have?
How many unique zeros does the function 
The zeros of a function are the x-coordinates at the points where the function (the y-coordinate, if you're graphing it) equals zero. So to solve for the zeros, set the function equal to zero:
Here you can then factor the common 
term, yielding:
Now you can factor the quadratic within the parentheses. This gives you:
The solutions to this equation are at the points 
and 
, demonstrating that there are 3 unique zeros to this function.
The zeros of a function are the x-coordinates at the points where the function (the y-coordinate, if you're graphing it) equals zero. So to solve for the zeros, set the function equal to zero:
Here you can then factor the common 
Now you can factor the quadratic within the parentheses. This gives you:
The solutions to this equation are at the points 
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What is the sum of all unique zeros to the function 
?
What is the sum of all unique zeros to the function 
The zeros of a function are the x-values at which the value of the function equals zero. So to solve for the zeros of a function, set the function equal to zero and then solve for 
. Here that means setting the function equal to zero:
Then factor the common 
:
Then factor the quadratic in parentheses:
Now make sure you solve for 
. The values of 
that lead to a product of zero are 
and 
. Therefore the sum is 
.
The zeros of a function are the x-values at which the value of the function equals zero. So to solve for the zeros of a function, set the function equal to zero and then solve for 
Then factor the common 
Then factor the quadratic in parentheses:
Now make sure you solve for 
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For the following system of equations, what shape does the corresponding graph make:
For the following system of equations, what shape does the corresponding graph make:
When we graph the three inequalities as equations, we get the below graph. Testing a point within the triangle does in fact fulfill all three conditions, so the shape formed by the inequalities is a triangle. If a tested point within the triangle did not fulfill one or more conditions, then we would have chosen “it does not form a shape.”
When we graph the three inequalities as equations, we get the below graph. Testing a point within the triangle does in fact fulfill all three conditions, so the shape formed by the inequalities is a triangle. If a tested point within the triangle did not fulfill one or more conditions, then we would have chosen “it does not form a shape.”
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Which of the following graphs correctly describes the system of inequalities:
Which of the following graphs correctly describes the system of inequalities:
After plotting the lines, shade the regions corresponding to each individual inequality. The area of intersection (in purple) is the solution to the system of inequalities. Another way to check your answer is to pick a point in all four regions delineated by the two equations and test those four coordinate points in the system of inequalities.
After plotting the lines, shade the regions corresponding to each individual inequality. The area of intersection (in purple) is the solution to the system of inequalities. Another way to check your answer is to pick a point in all four regions delineated by the two equations and test those four coordinate points in the system of inequalities.
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For the following system of equations, what shape does the corresponding graph make:
For the following system of equations, what shape does the corresponding graph make:
When we graph the three inequalities as equations, we get the below graph. Testing a point within the triangle does in fact fulfill all three conditions, so the shape formed by the inequalities is a triangle. If a tested point within the triangle did not fulfill one or more conditions, then we would have chosen “it does not form a shape.”
When we graph the three inequalities as equations, we get the below graph. Testing a point within the triangle does in fact fulfill all three conditions, so the shape formed by the inequalities is a triangle. If a tested point within the triangle did not fulfill one or more conditions, then we would have chosen “it does not form a shape.”
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Which of the following graphs correctly describes the system of inequalities:
Which of the following graphs correctly describes the system of inequalities:
After plotting the lines, shade the regions corresponding to each individual inequality. The area of intersection (in purple) is the solution to the system of inequalities. Another way to check your answer is to pick a point in all four regions delineated by the two equations and test those four coordinate points in the system of inequalities.
After plotting the lines, shade the regions corresponding to each individual inequality. The area of intersection (in purple) is the solution to the system of inequalities. Another way to check your answer is to pick a point in all four regions delineated by the two equations and test those four coordinate points in the system of inequalities.
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Which of the following ordered pairs (x,y) is a solution to the system of inequalities:
Which of the following ordered pairs (x,y) is a solution to the system of inequalities:
This is the only coordinate point that fulfills the system of inequalities.
This is the only coordinate point that fulfills the system of inequalities.
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If the system of inequalities 
and 
is graphed on the xy- plane above, which quadrant contains no solutions to the system?
If the system of inequalities 
The graph of the system of inequalities is shown below. The solution (in purple) does not go into region I or II.
The graph of the system of inequalities is shown below. The solution (in purple) does not go into region I or II.
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If the system of inequalities 
and 
is graphed on the xy- plane above, which quadrant contains no solutions to the system?
If the system of inequalities 
The graph of the system of inequalities is shown below. The solution (in purple) is present in all four regions.
The graph of the system of inequalities is shown below. The solution (in purple) is present in all four regions.
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If the system of inequalities 
and 
is graphed on the xy- plane above, which quadrant contains no solutions to the system?
If the system of inequalities 
The graph of the system of inequalities is shown below. The solution (in purple) is present in all four regions.
The graph of the system of inequalities is shown below. The solution (in purple) is present in all four regions.
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The graph shown above is best represented by which inequality?
The graph shown above is best represented by which inequality?
The x-intercept is (4,0) so when 
equals 0, 
, and the y-intercept is (0,3) so when 
equals 0, 
. These points only work for a line corresponding to 
. To determine which direction the inequality points, you can plug in any point. If we use (0,0), 
, so this is the direction the inequality must face.
The x-intercept is (4,0) so when 
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Which of the following are solutions to the system of equations 
and 
?
I) 
II) 
III) 
Which of the following are solutions to the system of equations 
I)
II)
III)
With this question, it is important to recognize the system of inequalities utilizes the “greater than” sign, not the “greater than or equal to” sign. 
does not fulfill 
. 
appears to fulfill both conditions but 
. 18 is not greater than 8, leaving only 
as a possible solution.
With this question, it is important to recognize the system of inequalities utilizes the “greater than” sign, not the “greater than or equal to” sign. 
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