The problem provides a quadratic function $f(x) = px^2 + 5x + q$ and its minimum point $(-2.5, -2.25)$. We are asked to: (a) Find the value of $p$, given that $p$ is an integer and $-2 < p < 2$. (b) Using the value of $p$ from (a), calculate $q$ and state the equation of the axis of symmetry. (c) Determine if the quadratic function changes if the graph is reflected in the x-axis and express the result in the form $f(x) = ax^2 + bx + c$.

1. Problem Description

The problem provides a quadratic function $f(x) = px^2 + 5x + q$ and its minimum point $(-2.5, -2.25)$. We are asked to:

(a) Find the value of $p$, given that $p$ is an integer and $-2 < p < 2$.

(b) Using the value of $p$ from (a), calculate $q$ and state the equation of the axis of symmetry.

(c) Determine if the quadratic function changes if the graph is reflected in the x-axis and express the result in the form $f(x) = ax^2 + bx + c$.

2. Solution Steps

(a) Finding the value of $p$:

Since $p$ is an integer and $-2 < p < 2$, the possible values for $p$ are $-1, 0, 1$. A quadratic function must have $p \neq 0$, and since the parabola opens upwards, $p > 0$. Therefore, $p = 1$.

(b) Finding the value of $q$ and the axis of symmetry:

We know that the vertex of the parabola is at $(-2.5, -2.25)$. The x-coordinate of the vertex is given by $x = -\frac{b}{2a}$, where $a = p$ and $b = 5$. In this case, $a = 1$.

So, $-2.5 = -\frac{5}{2(1)}$, which is true.

Since the vertex is $(-2.5, -2.25)$, we have $f(-2.5) = -2.25$.

$f(-2.5) = (1)(-2.5)^2 + 5(-2.5) + q = -2.25$

$6.25 - 12.5 + q = -2.25$

$-6.25 + q = -2.25$

$q = -2.25 + 6.25 = 4$

The axis of symmetry is a vertical line passing through the x-coordinate of the vertex, which is $x = -2.5$.

(c) Reflection in the x-axis:

When the graph of $f(x)$ is reflected in the x-axis, the new function becomes $-f(x)$.

If $f(x) = x^2 + 5x + 4$, then $-f(x) = -(x^2 + 5x + 4) = -x^2 - 5x - 4$.

Therefore, if the graph is reflected over the x-axis, the quadratic function becomes $f(x) = -x^2 - 5x - 4$.

3. Final Answer

(a) $p = 1$

(b) $q = 4$. The equation of the axis of symmetry is $x = -2.5$.

(c) Yes, the function changes. The reflected function is $f(x) = -x^2 - 5x - 4$.