help please!

Question

Consider the vertex form y= a(x - p)^2 + q of a quadratic function when answering the following question.

a) if a and q are both positive, then how many x- intercepts will there for this quadratic?explain

b) if p and q are both positive, then in which quadrant is the vertex? explain

c) if a is a negative number and q is a positive number, then how many x-intercepts will there for this quadratic. explain

d) write the equation of a quadratic function that opens down and has no x- intercepts. explain how you decided what values to use for each of the variables a,p and q.

Answer 1

The vertex form of parabola is y = a (x - p)²  + q

This is the standard form of vertical parabola.

Where (p, q) are vertex of the parabola.

It's general shape is upward U (or) downward U.

 

a) If a and q are positive

Since a is positive, then the parabola opens upward.

Since q is positive the y coordinate of vertex is positive.

So the graph of parabola is above x - axis.

There is no x intercepts.

Answer 2

b) If p and q are both positive,

p is x coordinate of vertex and q is the y coordinate of vertex

Therefore, the vertex in first quadrant.

Answer 3

c) If a is negative number and q is positive number

Since a is negative, the parabola opens downward.

Since q is positive the y coordinate of vertex is positive.

In this case the graph of parabola does not below the x - axis.

Therefore, the parabola(quadratic) has two x intercepts.

Answer 4

d) The quadratic equation is y = a (x - p)²  + q

We need equation of a quadratic function that opens down.

Since the parabola opens downward, a must be negative.

Required parabola has no x intercepts means the function does not cross the axis.

So the y coordinate(q) of vertex is not zero and it's already(down ward) it's not positive.

So far , If a and q are both negative, then the parabola opens down ward and has no x - intercepts.

There is no restrictions for p sign.

If p is positive then parabola in fourth quadrant.

If p is negative then the parabola in third quadrant.

Here i give two examples.