Quadratic Functions - Math
Card 0 of 76
Based on the figure below, which line depicts a quadratic function?
Based on the figure below, which line depicts a quadratic function?
Tap to see back →
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
Which of the following functions represents a parabola?
Which of the following functions represents a parabola?
Tap to see back →
A parabola is a curve that can be represented by a quadratic equation. The only quadratic here is represented by the function 
, while the others represent straight lines, circles, and other curves.
A parabola is a curve that can be represented by a quadratic equation. The only quadratic here is represented by the function
Find the center and radius of the circle defined by the equation:
Find the center and radius of the circle defined by the equation:
Tap to see back →
The equation of a circle is: 
where 
is the radius and 
is the center.
In this problem, the equation is already in the format required to determine center and radius. To find the 
-coordinate of the center, we must find the value of 
that makes 
equal to 0, which is 3. We do the same to find the y-coordinate of the center and find that 
. To find the radius we take the square root of the constant on the right side of the equation which is 6.
The equation of a circle is:
In this problem, the equation is already in the format required to determine center and radius. To find the
Find the center and radius of the circle defined by the equation:
Find the center and radius of the circle defined by the equation:
Tap to see back →
The equation of a circle is: 
where 
is the radius and 
is the center.
In this problem, the equation is already in the format required to determine center and radius. To find the 
-coordinate of the center, we must find the value of 
that makes 
equal to 
, which is 
. We do the same to find the y-coordinate of the center and find that 
. To find the radius we take the square root of the constant on the right side of the equation which is 10.
The equation of a circle is:
In this problem, the equation is already in the format required to determine center and radius. To find the
Find the 
-intercepts for the circle given by the equation:
Find the
Tap to see back →
To find the 
-intercepts (where the graph crosses the 
-axis), we must set 
. This gives us the equation:
Because the left side of the equation is squared, it will always give us a positive answer. Thus if we want to take the root of both sides, we must account for this by setting up two scenarios, one where the value inside of the parentheses is positive and one where it is negative. This gives us the equations:
and 
We can then solve these two equations to obtain 
.
To find the
Because the left side of the equation is squared, it will always give us a positive answer. Thus if we want to take the root of both sides, we must account for this by setting up two scenarios, one where the value inside of the parentheses is positive and one where it is negative. This gives us the equations:
We can then solve these two equations to obtain
Find the 
-intercepts for the circle given by the equation:
Find the
Tap to see back →
To find the 
-intercepts (where the graph crosses the 
-axis), we must set 
. This gives us the equation:
Because the left side of the equation is squared, it will always give us a positive answer. Thus if we want to take the root of both sides, we must account for this by setting up two scenarios, one where the value inside of the parentheses is positive and one where it is negative. This gives us the equations:
and 
We can then solve these two equations to obtain
To find the
Because the left side of the equation is squared, it will always give us a positive answer. Thus if we want to take the root of both sides, we must account for this by setting up two scenarios, one where the value inside of the parentheses is positive and one where it is negative. This gives us the equations:
We can then solve these two equations to obtain
Find the radius of the circle given by the equation:
Find the radius of the circle given by the equation:
Tap to see back →
To find the center or the radius of a circle, first put the equation in the standard form for a circle: 
, where 
is the radius and 
is the center.
From our equation, we see that it has not yet been factored, so we must do that now. We can use the formula 
.
, so 
.
and 
, so 
and 
.
Therefore, 
.
Because the constant, in this case 4, was not in the original equation, we need to add it to both sides:
Now we do the same for 
:
We can now find 
:
To find the center or the radius of a circle, first put the equation in the standard form for a circle:
From our equation, we see that it has not yet been factored, so we must do that now. We can use the formula
Therefore,
Because the constant, in this case 4, was not in the original equation, we need to add it to both sides:
Now we do the same for
We can now find
Find the center of the circle given by the equation:
Find the center of the circle given by the equation:
Tap to see back →
To find the center or the radius of a circle, first put the equation in standard form: 
, where 
is the radius and 
is the center.
From our equation, we see that it has not yet been factored, so we must do that now. We can use the formula 
.
, so 
.
and 
, so 
and 
.
This gives 
.
Because the constant, in this case 9, was not in the original equation, we must add it to both sides:
Now we do the same for 
:
We can now find the center: (3, -9)
To find the center or the radius of a circle, first put the equation in standard form:
From our equation, we see that it has not yet been factored, so we must do that now. We can use the formula
This gives
Because the constant, in this case 9, was not in the original equation, we must add it to both sides:
Now we do the same for
We can now find the center: (3, -9)
The graph of the equation
is a circle with what as the length of its radius?
The graph of the equation
is a circle with what as the length of its radius?
Tap to see back →
Rewrite the equation of the circle in standard form
as follows:
Since 
and 
, we complete the squares by adding:
The standard form of the equation sets
,
so the radius of the circle is
Rewrite the equation of the circle in standard form
as follows:
Since
The standard form of the equation sets
so the radius of the circle is
Determine the graph of the equation
Determine the graph of the equation
Tap to see back →
The equation of a circle in standard for is:
Where the center 
and the radius of the cirlce is 
.
Dividing by 4 on both sides of the equation yields
or
an equation whose graph is a circle, centered at (2,3) with radius = .5
The equation of a circle in standard for is:
Where the center
Dividing by 4 on both sides of the equation yields
or
an equation whose graph is a circle, centered at (2,3) with radius = .5
Give the radius and the center of the circle for the equation below.
Give the radius and the center of the circle for the equation below.
Tap to see back →
Look at the formula for the equation of a circle below.
Here 
is the center and 
is the radius. Notice that the subtraction in the center is part of the formula. Thus, looking at our equation it is clear that the center is 
and the radius squared is 
. When we square root this value we get that the radius must be 
.
Look at the formula for the equation of a circle below.
Here
Determine the equation of a circle whose center lies at the point 
and has a radius of 
.
Determine the equation of a circle whose center lies at the point
Tap to see back →
The equation for a circle with center 
and radius 
is :
Our circle is centered at 
with radius 
, so the equation for this circle is :
The equation for a circle with center
Our circle is centered at
What is the radius of the circle?
What is the radius of the circle?
Tap to see back →
The parent equation of a circle is represented by 
. The radius of the circle is equal to 
. The radius of the cirle is 
.
The parent equation of a circle is represented by
What is the center of the circle expressed by the funciton 
?
What is the center of the circle expressed by the funciton
Tap to see back →
The equation can be rewritten so that it looks like the parent equation for a circle 
. After completeing the square, the equation changes from 
to 
. From there it can be expressed as 
. Therefore the center of the circle is at 
.
The equation can be rewritten so that it looks like the parent equation for a circle
What are the coordinates of the center of a circle with the equation 
?
What are the coordinates of the center of a circle with the equation 
Tap to see back →
The equation of a circle is 
, in which (h, k) is the center of the circle. To derive the center of a circle from its equation, identify the constants immediately following x and y, and flip their signs. In the given equation, x is followed by -1 and y is followed by -6, so the coordinates of the center must be (1, 6).
The equation of a circle is 
What is the radius of a circle with the equation 
?
What is the radius of a circle with the equation 
Tap to see back →
To convert the given equation into the format 
, complete the square by adding 
to the x-terms and to the y-terms.
The square root of 4 is 2, so the radius of the circle is 2.
To convert the given equation into the format 
The square root of 4 is 2, so the radius of the circle is 2.
What is the radius of a circle with the equation 
?
What is the radius of a circle with the equation 
Tap to see back →
To convert the given equation into the format 
, complete the square by adding 
to the x-terms and to the y-terms.
The square root of 25 is 5, so the radius of the circle is 5.
To convert the given equation into the format 
The square root of 25 is 5, so the radius of the circle is 5.
Which equation does this graph represent?
Which equation does this graph represent?
Tap to see back →
The equation of a circle is 
, in which (h, k) is the center of the circle and r is its radius. Because the graph of the circle is centered at (0, 0), h and k are both 0. Because the radius is 3, the right side of the equation is equal to 9.
The equation of a circle is 
Which equation does this graph represent?
Which equation does this graph represent?
Tap to see back →
The equation of a circle is 
, in which (h, k) is the center of the circle and r is its radius. Because the graph of the circle is centered at (2, -3), h and k are -2 and 3. Because the radius is 4, the right side of the equation is equal to 16.
The equation of a circle is 
Based on the figure below, which line depicts a quadratic function?
Based on the figure below, which line depicts a quadratic function?
Tap to see back →
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