Tip #38 to Get a Top ACT Math Score (page 3)
When x is squared (x2) in an equation, the graph forms a U shaped curve called a parabola. The ACT usually shows the equation in standard form, y = ax2 + bx + c, or vertex form, y = a(x – h)2 + k. Either equation is called quadratic, which is just fancy vocab for having an x2 term.
In the equation y = ax2 + bx + c, the a tells whether the U-shaped graph opens up or down, and the c is the y intercept, the place where the graph crosses the y axis.
In the second equation y = a(x – h)2 + k, the h and k give the coordinates of the vertex of the graph (h,k), and the a again tells whether the U-shaped graph opens up or down. The vertex is the highest or lowest point of the graph and is therefore also called the maximum or minimum point.
This equation can also be written y – k = a(x – h)2, by just subtracting the k to the other side of the equation.
Let's look at this question:
If the graph of y = ax2 + bx + c is shown below, then the value of ac can be
F. positive only
G. negative only
H. zero only
J. positive or negative
K. positive, negative, or zero
Solution: Great question! From the diagram, we can tell that the U opens up, so the value of a must be positive, and the y intercept is negative, so c is negative. Thus, (a)(c) must be negative, since a positive number times a negative number equals a negative number: (+)(–) = (–).
Correct answer: G
- What are the coordinates for the y intercept in the graph of y = y2 + 2x – 3?
- (3, 0)
- (–3, 0)
- (0, 3)
- (0, –3)
- (0, 0)
- What are the coordinates for the maximum point in the graph of y – 2 = –(x + 3)2?
- (3, 2)
- (–3, 2)
- (3, –2)
- (–3, –2)
- (0, 0)
- The graph of y = ax2 + bx + c is shown below. When y = 0, x has
- 2 positive solutions
- 2 negative solutions
- 1 positive solution
- 1 negative solution
- 1 positive and 1 negative solution
- The graph of one of the following equations is the parabola shown below. Which one could it be?
- y – 3 = –(x – 2)2
- y + 3 = (x – 2)2
- y – 3 = –(x + 2)2
- y + 3 = –(x – 2)2
- y – 3 = (x – 2)2
- If y = kx2 + 3x + r is the equation for the parabola shown below, then the product of k and r is
- positive or negative
- D When the equation of a parabola is given in the form y = ax2 + bx + c, the c is the y-intercept. So for y = x2 + 2x –3, –3 is the y intercept, which has coordinates (0, –3), since the x value is 0 and the y value is –3.
- G Rewrite y – 2 = – (x – 3)2 so it matches the form y = (x – h)2 + k. Just add 2 to both sides to get y = – (x + 3)2 + 2. In this form, the vertex or minimum/maximum point is (h, k), so (–3, 2) is the maximum point.
- E Use the graph. When y = 0, the graph is at two different x values, one positive and one negative.
- F Our Skill makes this "hard" question easy! For the graph shown, the vertex clearly has a positive x and a positive y value, and the graph opens down. So we need h and k to be positive, and we need a negative sign in front of (x – 2)2. Answer choice F is the only equation that fits these requirements, y – 3 = – (x – 2)2.
- B Our Skill makes this "hard" question easy! From the graph, we can see that k must be positive, since the graph opens up. And we can see that r must be negative, since there is a negative y intercept. The product of k and r is (positive)(negative) = negative.
Go to: Tip #39
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