A hybrid Algebra-Geometry Problem

A parabola, whose equation is ax² + bx + 3 = y, has a vertex at (-1, -2). Find the value of a + b.

We know that any parabola can also be expressed in the form of a(x – c)² + d = y, where (c, d) is the point of the vertex of the parabola. In this example c = -1, and d = -2. Thus we have the equation in the vertex form as a(x –  (-1))² + (-2) = y. Setting this expression for y and the given expression for y equal, we have a(x + 1)² – 2 = ax² + bx + 3. Simplifying the left side, we have ax² + 2ax + a – 2 = ax² + bx + 3. Since the two sides are equal, we can compare their coefficients. The second and third terms provide with information we need, so we have 2a = b, and a – 2 = 3, which gives a = 5, and b = 2(5) = 10. Thus a + b = 5 + 10 = 15.

The second way of solving this problem is to remember that the x-coordinate of the vertex of a parabola is given by -b/2a, so that -1 = -b/2a, so that 2a = b. Now, we can plug in the x-coordinate of the given vertex point into the given equation to get a second expression for constants a and b: a(-1)² + b(-1) + 3 = -2, which gives a – b = -5. Substituting 2a = b into this equation, we have a – 2a = -5, so that a = 5, and b = 10 as before. 

Always check the constants you found with the vertex coordinate in the original equation to see if you are correct. If you are not sure how to use either of the two methods, skip this question on the exam and come back to it.  

_________________________________________________________

Please visit our website RedesignedSATMath.com to find the most effective prep eBooks currently on sale for the New SAT Exam as well as online tutoring information.

Our online Math Prep system contains 1,200+ Math problems (practice questions and complete SAT Math Practice Tests). 

Gain Access Now by using the PayPal link: https://www.paypal.com/cgi-bin/webscr?cmd=_s-xclick&hosted_button_id=SWRYVDKLJYGYG

You will receive the username and password to your online prep account after your PayPal payment is confirmed