Question

Parabola properties Consider the general quadratic function \(f(x)=a x^{2}+b x+c,\) with \(a \neq 0\). a. Find the coordinates of the vertex in terms of \(a\). \(b\), and \(c\). b. Find the conditions on \(a, b,\) and \(c\) that guarantee that the graph of \(f\) crosses the \(x\) -axis twice.

Short answer

Answer: The vertex of a quadratic function, given by \(f(x) = ax^2 + bx + c\), has coordinates \((h, k) = \left(-\frac{b}{2a}, -\frac{b^2}{4a} + c\right)\). For the function to cross the x-axis twice, the discriminant must be positive: \(\Delta > 0\) where \(\Delta = b^2 - 4ac\).

Step-by-step solution

(Step 1: Part a – Find the coordinates of the vertex of the parabolic function)

Finding the vertex of the parabola requires completing the square. Given the quadratic function \(f(x) = ax^2 + bx + c\), first factor out the leading coefficient \(a\): $$ f(x) = a(x^2 + \frac{b}{a}x) + c $$ Next, we will use the rule of completing the square which states that \((x + \frac{b}{2a})^2\) is the square of a binomial when the original quadratic is \(x^2 + \frac{b}{a}x\) plus a constant that has to be added or subtracted for it to be perfectly square trinomial. So, we can rewrite \(f(x)\) as: $$ f(x) = a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} - \frac{b^2}{4a^2}\right) + c $$ Now, we can form the perfect square trinomial with \((x + \frac{b}{2a})^2\): $$ f(x) = a\left(\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}\right) + c $$ The vertex of the parabola is in the form \((h, k)\), where \(h = -\frac{b}{2a}\) determines the x-coordinate, and \(k\) represents the minimum or maximum value of the function: $$ h = -\frac{b}{2a} $$ Substitute the expression of \(h\) back to the function \(f(x)\) to find the value of \(k\): $$ k = f(h) = a\left(\left(-\frac{b}{2a} + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}\right) + c = a\cdot\left( - \frac{b^2}{4a^2}\right) + c = -\frac{b^2}{4a} + c $$ Thus, the vertex is given by \((h, k) = \left(-\frac{b}{2a}, -\frac{b^2}{4a} + c\right)\).

(Step 2: Part b - Find the conditions for f to cross the x-axis twice)

To find the conditions for the parabola to cross the x-axis twice, we need to determine when there are two distinct real roots of the quadratic equation. We can achieve this by calculating the discriminant and setting constraints. Recall that the discriminant is given by the formula: $$ \Delta = b^2 - 4ac $$ For a quadratic function to cross the x-axis twice, the discriminant must be positive because of the quadratic formula: $$ x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a} $$ If the discriminant is positive, it means that there are two distinct real roots. Thus, the condition required is: $$ \Delta > 0 \implies b^2 - 4ac > 0 $$

Key concepts

Vertex of a Parabola

The vertex is a significant point on a parabola. It represents the highest or lowest point, depending on whether the parabola opens upwards or downwards. In the context of a general quadratic function given by \(f(x) = ax^2 + bx + c\), determining the vertex involves a specific process.

The x-coordinate of the vertex, often denoted as \(h\), can be found using the formula:

• \(h = -\frac{b}{2a}\)

To find the y-coordinate, substitute \(h\) back into the function:

\(k = f(h) = a\left(\left(-\frac{b}{2a}\right)^2\right) - \frac{b^2}{4a} + c\)

Simplifying further leads us to the full coordinate of the vertex:

• \((h, k) = \left(-\frac{b}{2a}, -\frac{b^2}{4a} + c\right)\)

This point shows the location where the parabola changes direction, and understanding it is crucial for graphing quadratic functions.

Discriminant in Quadratic Equations

The discriminant is a component of the quadratic formula critical for determining the nature of the roots of a quadratic equation. For the equation \(ax^2 + bx + c=0\), the discriminant is expressed as \(\Delta = b^2 - 4ac\).

The value of the discriminant provides clear insights:

• **\(\Delta > 0\):** There are two distinct real roots. This means the parabola intersects the x-axis at two different points.
• **\(\Delta = 0\):** There is exactly one real root, indicating the parabola touches the x-axis at just one point, also known as the vertex.
• **\(\Delta < 0\):** No real roots exist, which implies the parabola does not intersect the x-axis at any point.

Understanding the discriminant is fundamental when analyzing a quadratic equation’s ability to intersect the x-axis.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a form that reveals the vertex of the parabola directly. It involves rewriting the quadratic function to highlight the perfect square trinomial.

Take a quadratic function of the form \(f(x) = ax^2 + bx + c\). Here is the step-by-step approach to complete the square:

• Start by isolating the quadratic and linear terms: factor out \(a\):
• \(f(x) = a(x^2 + \frac{b}{a}x) + c\)
• To complete the square, add and subtract \(\left(\frac{b}{2a}\right)^2\) inside the parenthesis:
• \(f(x) = a(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} - \frac{b^2}{4a^2}) + c\)
• Reformulate to form a square: \(a((x + \frac{b}{2a})^2 - \frac{b^2}{4a^2}) + c\)

With these steps, you can find a form where the expression \((x + \frac{b}{2a})^2\) clearly states the vertex of the parabola. Completing the square is part of what makes quadratic equations readily analysable.

Real Roots of Quadratic Equations

Quadratic equations, written as \(ax^2 + bx + c = 0\), can have up to two roots, which are solutions of the equation. Whether these roots are real or complex depends largely on the discriminant \(\Delta = b^2 - 4ac\).

For real roots to exist, the discriminant must be non-negative, as follows:

• **Two distinct real roots:** If \(\Delta > 0\) \((b^2 - 4ac >0)\), indicating the parabola crosses the x-axis twice.
• **One real root (repeated):** If \(\Delta = 0\), meaning the parabola touches the x-axis at a single point (vertex).
• **No real roots:** If \(\Delta < 0\), which refers to complex roots, implying the parabola does not intersect the x-axis at all.

Recognizing and calculating these roots is vital for solving quadratic equations. Understanding their nature provides insight into the behavior of the graph of the function.