Determine the interval(s) on which the function is increasing and decreasing. g(x)= 5(x+3)^2-2

Answer 1

The vertex of this parabola has an x-value of -3. The (x+3)² factor has +3, just change the sign to get the the x value of the vertex.

Since this is an upward opening parabola, that means x=-3 is the line of symmetry, with the parabola increasing for x>-3 and decreasing for x<-3

You know it's upward opening because the coefficient 5 is positive. Had it been negative, it would have been a downward opening parabola with the opposite intervals increasing and decreasing.

You know it's a parabola because the two variables x and y have one linear and one quadratic.

When y is linear, that's either an upward opening or downward opening parabola

Had it been reversed, with x linear and y quadratic, you would have a left or right opening parabola.

g(x)=5(x+3)²-2 could be differentiated, after expanding to g(x)=5x²+30x+45-2=5x²+30x+43

g'(x)=10x+30 set this = 0, solve for x

10x+30=0

10x=-30

x=-30/10=-3 This is the point that separates the increasing or decreasing intervals. At the point x=-3 it's neither increasing or decreasing.

To take that derivative, you multiply a coefficient of an x term by it's power and reduce the power by 1

derivative of 5x² is 10x (5 times 2) reduce the power of 2 by 1 to get just x

derivative of 30x is 30 (30 times 1) and reduce the power to 0. x⁰=1 anything to the zero power =1

derivative of the constant term is always zero

leaving you with g'(x)=10x+30 the ' after the g indicates a derivative

Answer 2

Hi there!

Since this is tagged as precalculus, I'm going to assume you're not using derivatives to solve this quite yet.

Instead, we can determine the intervals on which g(x) is increasing by realizing that this function is written in vertex form for a parabola: f(x) = a(x – h)² + k. With this form, we know easily that the vertex is located at (h, k) – in the case of g, the vertex is located at (-3, -2). Since this parabola is face-up, we also know that it's decreasing to the left of the vertex and increasing to the right. Hence, we can solve that g(x) is increasing from x = -3 to x = ∞. In interval notation, it's increasing on (-3, ∞).

I hope this helped, and let me know if you have any questions!