Other Important Graphs and Equations
Other Important Graphs and Equations
In addition to the graphs and equations of lines, the Math IIC will test your understanding of the graphs and equations of parabolas, circles, ellipses, and hyperbolas.
Questions on these topics either will ask you to match up the correct graph with the correct equation or give you an equation and ask you to figure out certain characteristics of the graph.
Most of the questions about conic sections are straightforward. If you know the information in the sections below, you’ll be able to breeze through them.
Parabolas
A parabola is a U-shaped curve that can open either upward or downward.
A parabola is the graph of a quadratic function, which, you may recall, is ax2 + bx + c. The equation of a parabola can take on two different forms—the standard form and the general form. Each can help you determine different information about the nature of the parabola.
Standard Form of the Equation of a Parabola
The standard form of the equation of a parabola is perhaps the most useful and will be the one most used on the Math IIC test:
where a, h, and k are constants. From this formula, you can learn a few pieces of information:
  1. The vertex of the parabola is (h, k).
  2. The axis of symmetry of the parabola is the line x = h.
  3. The parabola opens upward if a > 0 and downward if a < 0.
For example, if you were given the parabola equation y = –3(x – 5)2 + 8, you first need to pick out the values of the constants a, h, and k. Then you can derive information about the parabola. For this example, a = –3, h = 5, and k = 8. So the vertex is (5, 8), the axis of symmetry is the line x = 5, and since –3 < 0, the parabola opens downward.
General Form of the Equation of a Parabola
The general form of the equation of a parabola is:
where a, b, and c are constants. If a question presents you with a parabola equation in this form, you can find out the following information about the parabola:
  1. The vertex of the parabola is (b /2a, c b/4a).
  2. The axis of symmetry of the parabola is the line x = –b/ 2a.
  3. The parabola opens upward if a > 0 and downward if a < 0.
  4. The y-intercept is the point (0, c).
Circles
A circle is the collection of points equidistant from a given point, called the center of the circle. For the Math IIC test, there is only one equation you have to know for a circle. This equation is called the standard form:
where (h, k) is the center of the circle and r is the radius. When the circle is centered at the origin, so that h = k = 0, then the equation simplifies to:
That’s it. That’s all you need to know about a circle in coordinate geometry. Once you know and understand this equation, you should be able to sketch a circle in its proper place on the coordinate system if given its equation. You will also be asked to figure out the equation of a circle given a picture of its graph.
To see if you know what you need to know, try to answer the following practice problem:
What is the equation of the circle pictured below?
The center is given in the image: (–2 ,–1). All you need to finish the formula is the radius. We do this by finding the distance from the center and the point (2, –4) pictured on the circle:
The radius of the circle is 5, so the equation of the circle can be written as (x + 2)2 + (y + 1)2 = 25.
Ellipses
An ellipse is a figure shaped like an oval. It looks like a circle somebody sat on, but it’s a good deal more complicated than a circle, as you can see from the diagram below.
An ellipse is formed by a set of points at a constant summed distance from two fixed points called the foci. The line segment containing the foci of an ellipse with both endpoints on the ellipse is called the major axis. The endpoints of the major axis are called the vertices. The line segment perpendicular to the major axis with both endpoints on the ellipse is the minor axis. The point halfway between the foci is the center of the ellipse. When you see an ellipse, you should be able to identify where each of these components would be.
The two foci are crucial to the definition of an ellipse. The sum of the distances from both foci to any point on the ellipse is constant. For every point on the ellipse, the cumulative distance from the two foci to that point will be constant. In the image below, for example, d1 + d2 is equal to d3 + d4.
The standard form of the equation of an ellipse is:
where a, b, h, and k are constants. With respect to this formula, remember that:
  1. The center of the ellipse is (h, k).
  2. The length of the horizontal axis is 2a.
  3. The length of the vertical axis is 2b.
  4. If a > b, the major axis is horizontal and the minor axis is vertical; if b > a, the major axis is vertical and the minor axis is horizontal.
When an ellipse is centered at the origin so that h = k = 0, the standard form of the equation of an ellipse becomes:
On the test, you might see a question like this:
What are the coordinates of the center and vertices of an ellipse given by the following equation?
First, find the center of the ellipse. By comparing this equation to the standard form, you see that (h, k) = (2, –5). Since the vertices are the endpoints of the major axis, your next step should be to find the orientation and length of that axis. In this ellipse, b > a, so the major axis is vertical and is 2b = 2 = 12 units long. The coordinates of the vertices are therefore (2, –5 ± 6), which works out to (2, –11) and (2, 1).
Hyperbolas
Though hyperbolas appear infrequently on the Math IIC, you should still review them.
A hyperbola is shaped like two parabolas facing away from each other:
The two parts of a hyperbola can open upward and downward, like they do in the previous graph, or they can open to the sides, like the hyperbola below:
The standard form of the equation of a hyperbola that opens to the sides is:
where a, b, h, and k are constants. The standard form of the equation of a hyperbola that opens upward and downward is the same as the side form except that the (xh) and (yk) terms are interchanged:
The center of a hyperbola is (h, k), and the axis of symmetry is the line x = h for vertical hyperbolas or y = k for horizontal hyperbolas.
Similar to the equations for a circle and ellipse, the equation of a hyperbola becomes simpler for a hyperbola centered at the origin:
If you see a question about a hyperbola, it will most likely concern the center of the hyperbola, which can be readily found using the equation of the hyperbola.
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