Graph the parabola

Question

Graph the y square = -16(x+3). Describe the characteristics of the graph.

Answer 1

You'll have to graph it yourself, but we can tell the characteristics of the graph just by looking at the equation.

 

To do this, you'll want to convert equation graph to "vertex" form.  Vertex form is generally written like this:

 

y = a(x - h)² + k    ...or...   x = a(y - k)² + h

 

...depending on whether the parabola is a function of x or a function of y.

 

Your equation LOOKS like one of these but actually isn't.  The equation needs to be written as a function of one of the variables, but the square term and the non-square term are reversed from what they would normally be, so it is not the way we need it.  We have to convert it:

 

y² = -16 (x + 3)

y² = -16x - 48

16x = -y² - 48

x = (-1/16) y² - 3.

 

So the equation is of the form x = a(y - k)2 + h.

 

That tells us that x is a function of y, which means that the parabola is horizontal and opens to the left or right.

 

So the first thing we can tell is from the value of a, which in this case is -1/16.  The coefficient a indicates two things about the parabola. The sign of a tells whether the parabola opens to the right (positive a) or to the left (negative a). Your a is negative, so it opens to the left.

Second, the absolute value of a tells whether the parabola is skinny or fat. The larger the value, the skinnier the parabola. 1/16 is less than 1, much closer to 0 than 1, in fact, so that means that you have a relatively have a relatively fat parabola.

 

Finally, the point (h, k) is the vertex of the parabola. In your case, you just have y², not (y ± something)².  That means that k is 0.  And the -3 at the end means that h is -3.  So the vertex is at (-3, 0).