In addition to the graphs and equations of lines, the Math IC will testyour understanding of the graphs and equations of parabolas andcircles.
Questions on these topics will either askyou to match the correct graph with the correct equation or give you anequation and ask you to figure out certain characteristics of the graph.
Most of the questions about parabolas andcircles are straightforward. If you know the information in thesections 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 be expressed in two forms—the standardform and the general form. Each can help you determine differentinformation about the nature of the parabola.
Standard Form of the Equation of a Parabola
The standard form of the equation of aparabola is perhaps the most useful and will be the one most used onthe Math IC test:
where
a,
h, and
k are constants. From this formula, you can learn a few pieces of information:
- The vertex of the parabola is (h, k).
- The axis of symmetry of the parabola is the line x = h.
- 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 areconstants. If a question presents you with a parabola equation in thisform, you can find the following information about the parabola:
- The vertex of the parabola is (–b /2a, c – b
/4a). - The axis of symmetry of the parabola is the line x = –b/ 2a.
- The parabola opens upward if a > 0, and downward if a < 0.
- The y-intercept is the point (0, c).
Circles
A
circle is the collection of pointsequidistant from a given point, called the center of the circle. Forthe Math IC test, there is only one equation you have to know for acircle. 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 abouta circle in coordinate geometry. Once you know and understand thisequation, you should be able to sketch a circle in its proper place onthe coordinate system if you are given its equation. You will also beasked to figure out the equation of a circle if you are given a pictureof its graph.
To test your knowledge, try to answer the following practice problem:
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| What is the equation of the circle pictured below? |
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The center is given in the image: (–2, –1).All you need to finish the formula is the radius. We determine this byfinding the distance from the center and the point, (2, –4), picturedon 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.
