Question

P (x) = a (x') + c, where a < 0 and c > 0. Which TWO graphs could represent p(x)?

Answer 1

The two graphs that could represent p(x) are options B and A respectively.

How is it so?

A parabola that opens downwards is a type of quadratic function. The general form of a quadratic function is given by:

`\[ f(x) = ax^2 + bx + c \]`

For a parabola that opens downwards, the coefficient
`\(a\)` must be negative
`(\(a < 0\))`. The vertex form of the quadratic function is also helpful in understanding the properties of the parabola:

`\[ f(x) = a(x - h)^2 + k \]`

In this form, the vertex of the parabola is the point
`\((h, k)\)`. When
`\(a < 0\)`, the parabola opens downwards.

Since the value of a is less than 0, the graph of the parabola would be opening downwards. Because of this we can rule out option C. In a quadratic equation, c represents the y-intercept, and, in this case c is negative, meaning the y-intercept is less than 0 and where c is positive, the y-intercept is positive.

Only option B has a downward-opening curve and a y-intercept less than 0 and option A has a downward-opening curve and a y-intercept greater than 0, so, it is both options B for where a is less than 0 and option A where c is greater than 0.