Conic sections on the ACT Mathematics test cover the graphs and standard equations of circles, ellipses, parabolas, and hyperbolas in the coordinate plane.
A student must read an equation in standard form and pull out features such as the center, vertices, foci, directrices, and the slopes of a hyperbola's asymptotes. Items usually give a completed-square equation and ask which choice lists those features, the coordinates of the foci, or an equation of a parabola's directrix.
Common traps include swapping a and b, mixing horizontal and vertical openings, and treating a translated graph as if it were centered at the origin.
Sample questions
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Question 1Mid
Which of the following gives the center, vertices, and asymptote slopes of 16(x+1)2−9(y−3)2=1?
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Answer: D — Center (−1,3); vertices (−5,3) and (3,3); slopes ±43
The center is (−1,3), and the positive x term gives a horizontal transverse axis with a=4. Thus, the vertices are (−5,3) and (3,3) and the asymptote slopes are ±b/a=±43.
Question 2Mid
The parabola y2=12x is graphed in the standard (x,y) coordinate plane. Which of the following is an equation of its directrix?
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Answer: A — x=−3
For y2=4px with 4p=12,p=3. The parabola has vertex (0,0) and opens right, so its directrix is the vertical line x=−3.
Question 3Mid
In the standard (x,y) coordinate plane, what are the coordinates of the foci of the hyperbola 16x2−9y2=1?
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Answer: A — (±5,0)
For 16x2−9y2=1,c2=a2+b2=16+9=25, so c=5. The transverse axis is horizontal, so the foci are (±5,0).
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