Without solving, determine the number of roots for this equation, y = −4(x − 7)^2.

Question

Explain and sketch the graph please.

Answer 1

7 is a root of multiplicity 2

axis of symmetry at x=7

vertex at (7,0)

y-intercept at (0,-196)

Answer 2

There is one root because of the (x-7) term. A root is when the function crosses the x value, that is when y = 0, so if we were to set this equal to 0, only one x value could give us y = 0 and that is x = 7.

We also know this is a parabola because of the squared and it is concave down, like a rainbow, because of the negative constant value (-4 here).

Answer 3

Hi Carlos,

First, this is a quadratic equation in what we call Vertex Form. We know this because it follows the form:

Where the vertex, or the "point" of the parabola, is given by the coordinate (h,k). In this case, it is at the point (7,0).

A quadratic equation can have zero, one, or two solutions, depending how many times it touches the x-axis. Since there is no k term in this problem, you know that the vertex is on the x-axis, meaning it has one solution, or root.

We can graph by plugging in various numbers for x, such as -2, -1, 0, 1, 2, etc. and plotting each point, but we can also figure out what the graph would look like through intuition. Since the coefficient out front is negative, the parabola would flip upside down. The coefficient being 4 will also cause the graph to be narrowed by a factor of 4.

Hope this helps!