Quadratic Inequalities - Algebra II
Card 0 of 20
Find an inequality for points on a graph that fall on or inside of a circle centered at 
with a radius of 
, as shown below.
Find an inequality for points on a graph that fall on or inside of a circle centered at
The equation for a circle centered at point 
with radius 
is 
. Our circle is centered at 
with 
, and we are interested in points that lie along or inside of the circle. This means the left-hand side must be less than or equal to the right-hand side of the equation. We are left with 
or
The equation for a circle centered at point
Compare your answer with the correct one above
Given the above circle inequality, which point is not on the edge of the circle?
Given the above circle inequality, which point is not on the edge of the circle?
This is a graph of a circle with radius of 5 and a center of (1,1). The center of the circle is not on the edge of the circle, so that is the correct answer. All other points are exactly 5 units away from the circle's center, making them a part of the circle.
This is a graph of a circle with radius of 5 and a center of (1,1). The center of the circle is not on the edge of the circle, so that is the correct answer. All other points are exactly 5 units away from the circle's center, making them a part of the circle.
Compare your answer with the correct one above
Given the above circle inequality, which point satisfies the inequality?
Given the above circle inequality, which point satisfies the inequality?
The left side of the equation must be greater than or equal to 25 in order to satisfy the equation, so plugging in each of the values for x and y, we see that:
The only point that satisfies the inequality is (7,4) since it yields an answer that is greater than or equal to 25.
The left side of the equation must be greater than or equal to 25 in order to satisfy the equation, so plugging in each of the values for x and y, we see that:
The only point that satisfies the inequality is (7,4) since it yields an answer that is greater than or equal to 25.
Compare your answer with the correct one above
Given the above circle inequality, does the center satisfy the equation?
Given the above circle inequality, does the center satisfy the equation?
The center of the circle is 
, so plugging those values in for x and y yields the response that 0 is greater than or equal to 25.
Since plugging in the center values gives us a false statement we know that our center does not satisfy the inequality.
The center of the circle is
Since plugging in the center values gives us a false statement we know that our center does not satisfy the inequality.
Compare your answer with the correct one above
Given the above circle inequality, is the shading on the graph inside or outside the circle?
Given the above circle inequality, is the shading on the graph inside or outside the circle?
Check the center of the circle to see if that point satisfies the inequality.
When evaluating the function at the center (1,1), we see that it does not satisfy the equation, so it cannot be in the shaded region of the graph.
Therefore the shading is outside of the circle.
Check the center of the circle to see if that point satisfies the inequality.
When evaluating the function at the center (1,1), we see that it does not satisfy the equation, so it cannot be in the shaded region of the graph.
Therefore the shading is outside of the circle.
Compare your answer with the correct one above
Given the above circle inequality, which point is not on the edge of the circle?
Given the above circle inequality, which point is not on the edge of the circle?
This is a graph of a circle with radius of 6 and a center of (-2,4). The point (2,2) is not on the edge of the circle, so that is the correct answer. All other points are exactly 6 units away from the circle's center, making them a part of the circle.
This is a graph of a circle with radius of 6 and a center of (-2,4). The point (2,2) is not on the edge of the circle, so that is the correct answer. All other points are exactly 6 units away from the circle's center, making them a part of the circle.
Compare your answer with the correct one above
Given the above circle inequality, which point satisfies the inequality?
Given the above circle inequality, which point satisfies the inequality?
The left side of the equation must be greater than or equal to 36 in order to satisfy the equation, so plugging in each of the values for x and y, we see:
Therefore only 
yields an answer that is greater than or equal to 36.
The left side of the equation must be greater than or equal to 36 in order to satisfy the equation, so plugging in each of the values for x and y, we see:
Therefore only
Compare your answer with the correct one above
Given the above circle inequality, does the center satisfy the equation?
Given the above circle inequality, does the center satisfy the equation?
The center of the circle is 
, so plugging those values in for x and y yields the response,
Therefore, the center does not satisfy the inequality.
The center of the circle is
Therefore, the center does not satisfy the inequality.
Compare your answer with the correct one above
Given the above circle inequality, is the shading on the graph inside or outside the circle?
Given the above circle inequality, is the shading on the graph inside or outside the circle?
Check the center of the circle to see if that point satisfies the inequality. When evaluating the function at the center (-2,4), we see that it does not satisfy the equation, so it cannot be in the shaded region of the graph. Therefore the shading is outside of the circle.
Check the center of the circle to see if that point satisfies the inequality. When evaluating the function at the center (-2,4), we see that it does not satisfy the equation, so it cannot be in the shaded region of the graph. Therefore the shading is outside of the circle.
Compare your answer with the correct one above
Given the above circle inequality, which point is not on the edge of the circle?
Given the above circle inequality, which point is not on the edge of the circle?
Recall the equation of a circle:
where r is the radius and (h,k) is the center of the circle.
This is a graph of a circle with radius of 2 and a center of (-4,-3). The point (2,3) is not on the edge of the circle, so that is the correct answer.
All other points are exactly 2 units away from the circle's center, making them a part of the circle's edge.
Recall the equation of a circle:
This is a graph of a circle with radius of 2 and a center of (-4,-3). The point (2,3) is not on the edge of the circle, so that is the correct answer.
All other points are exactly 2 units away from the circle's center, making them a part of the circle's edge.
Compare your answer with the correct one above
Given the above circle inequality, which point satisfies the inequality?
Given the above circle inequality, which point satisfies the inequality?
The left side of the equation must be less than or equal to 4 in order to satisfy the equation, so plugging in each of the values for x and y, we see:
The only point that satisfies the inequality is the point (-3,-2), since it yields an answer that is less than or equal to 4.
The left side of the equation must be less than or equal to 4 in order to satisfy the equation, so plugging in each of the values for x and y, we see:
The only point that satisfies the inequality is the point (-3,-2), since it yields an answer that is less than or equal to 4.
Compare your answer with the correct one above
Given the above circle inequality, does the center satisfy the equation?
Given the above circle inequality, does the center satisfy the equation?
Recall the equation of circle:
where r is the radius and the center of the circle is at (h,k).
The center of the circle is (-4,-3), so plugging those values in for x and y yields the response that 0 is less than or equal to 4, which is a true statement, so the center does satisfy the inequality.
Recall the equation of circle:
The center of the circle is (-4,-3), so plugging those values in for x and y yields the response that 0 is less than or equal to 4, which is a true statement, so the center does satisfy the inequality.
Compare your answer with the correct one above
Given the above circle inequality, is the shading on the graph inside or outside the circle?
Given the above circle inequality, is the shading on the graph inside or outside the circle?
Check the center of the circle to see if that point satisfies the inequality. When evaluating the function at the center (-4,-3), we see that it does satisfy the equation, so it can be in the shaded region of the graph. Therefore the shading is inside of the circle.
Check the center of the circle to see if that point satisfies the inequality. When evaluating the function at the center (-4,-3), we see that it does satisfy the equation, so it can be in the shaded region of the graph. Therefore the shading is inside of the circle.
Compare your answer with the correct one above
What is the 
-intercept of 
?
What is the
The 
-intercepts of a function are the points where 
. When we substitute this into our equation, we get:
.
Adding nine to both sides,
.
Modifying the equation to get like bases get us,
Since 
.
Now we can set the exponents equal to eachother and solve for 
.
Thus,
.
Giving us our final solution:
.
The
Adding nine to both sides,
Modifying the equation to get like bases get us,
Since
Now we can set the exponents equal to eachother and solve for
Thus,
Giving us our final solution:
Compare your answer with the correct one above
Which equation would match to this graph:
Which equation would match to this graph:
The general equation for a circle is 
where the center is 
and its radius is 
.
In this case, the center is 
and the radius is 
, so the equation for the circle is 
.
We can simplify this equation to: 
.
The circle is shaded on the inside, which means that choosing any point 
and plugging it in for 
would produce something less than 
.
Therefore, our answer is 
.
The general equation for a circle is
In this case, the center is
We can simplify this equation to:
The circle is shaded on the inside, which means that choosing any point
Therefore, our answer is
Compare your answer with the correct one above
Which equation would produce this graph:
Which equation would produce this graph:
The general equation of a circle is 
where the center is 
and the radius is 
.
In this case, the center is 
and the radius is 
, so the equation for this circle is 
.
The circle is shaded on the inside, which means that choosing any point 
and plugging it in for 
would produce something less than 
.
Therefore, our answer is 
.
The general equation of a circle is
In this case, the center is
The circle is shaded on the inside, which means that choosing any point
Therefore, our answer is
Compare your answer with the correct one above
Which inequality does this graph represent?
Which inequality does this graph represent?
The hyperbola in the question's graph is 
.
This could be discerned because it subtracts 
from 
and not the other way around, putting the zeros on the 
axis. This narrows the answers down to 
and 
.
Test a point to tell which inequality is being used in the graph.
The point 
is within the shaded region, for example.
, which is greater than 16, so the answer is 
.
The hyperbola in the question's graph is
This could be discerned because it subtracts
Test a point to tell which inequality is being used in the graph.
The point
Compare your answer with the correct one above
Which inequality does this graph represent?
Which inequality does this graph represent?
The hyperbola in the graph has y-intercepts rather than x-intercepts, so the equation must be in the form 
and not the other way around.
The y-intercepts are at 1 and -1, so the correct equation will have just 
and not 
.
The answer not must either be,
or 
.
To see which, test a point in the shaded area.
For example, 
.
, which is less than 1, so the answer is 
.
The hyperbola in the graph has y-intercepts rather than x-intercepts, so the equation must be in the form
The y-intercepts are at 1 and -1, so the correct equation will have just
The answer not must either be,
To see which, test a point in the shaded area.
For example,
Compare your answer with the correct one above
Which of the following inequalities is not hyperbolic?
Which of the following inequalities is not hyperbolic?
The equation for a horizontal hyperbola is
. The equation for a vertical hyperbola is 
. Hyperbolic inequalities use an inequality sign rather than an equals sign, but otherwise have the same form as hyperbolic equations. The fact that the right side of the inequality is not equal to 1 does not change the fact that 
, 
and 
represent hyperbolas, since THESE can all be simplified to create an inequality with 1 on the right side (by dividing both sides of the equation by the constant on the right side of the inequality.) Answer choice 
is the only option in which the two terms on the left side of the inequality are combined using addition rather than subtraction, creating an ellipse rather than a hyperbola. (The equation for an ellipse is 
.)
The equation for a horizontal hyperbola is
Compare your answer with the correct one above
Which of the following inequalities is not hyperbolic?
Which of the following inequalities is not hyperbolic?
The equation for a horizontal hyperbola is 
. The equation for a vertical hyperbola is 
. Hyperbolic inequalities use an inequality sign rather than an equals sign, but otherwise have the same form as hyperbolic equations. The presence of coefficients in 
and 
does not change the fact that 
and 
represent hyperbolas, since both can be simplified to remove those coefficients (by dividing the numerator and denominator of terms with coefficients by those coefficients.) Answer choice 
is missing an exponent of 2 on the first term in the inequality, and therefore does not match the form of a hyperbola.
The equation for a horizontal hyperbola is 
Compare your answer with the correct one above