Question

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

Solution: The vertex coordinates of the quadratic function are: \(h = -\frac{b}{2a}\) and \(k = c - \frac{b^2}{4a}\). The conditions on a, b, and c to guarantee that the graph crosses the x-axis twice are: \(b^2 - 4ac > 0\).

Step-by-step solution

Complete the square

In order to find the vertex of the quadratic, the given quadratic function in the form \(f(x)=ax^2+bx+c\) should be rewritten in the vertex form, which is \(f(x)=a(x-h)^2+k\), where (h, k) is the vertex of the quadratic. To rewrite the quadratic in vertex form, we need to complete the square. Consider the quadratic function \(f(x) = ax^2 + bx + c\). We need to rewrite this function in the form \(f(x)=a(x-h)^2+k\). To complete the square, we perform the following steps: 1) Divide the coefficient of the linear term (b) by 2, which is \(\frac{b}{2}\) 2) Square the result to get \(\left(\frac{b}{2}\right)^2\), which is \(\frac{b^2}{4}\) 3) Add and subtract this value within the parenthesis in the form \(ax^2+bx+\frac{b^2}{4}-\frac{b^2}{4}+c\) Now rewrite the quadratic as: $$f(x)=a\left(x^2+\frac{b}{a}x+\frac{b^2}{4a^2}\right)-\frac{b^2}{4a}+c$$ Factor out a from the first term: $$f(x)=a\left(x+\frac{b}{2a}\right)^2-\frac{b^2}{4a}+c$$ Now our quadratic is in vertex form, with \(h=-\frac{b}{2a}\) and \(k=c-\frac{b^2}{4a}\).

Find the vertex coordinates

Now that we have our quadratic function in vertex form, we can easily find the coordinates of the vertex (h, k) in terms of a, b, and c. The vertex coordinates are: $$h=-\frac{b}{2a}$$ $$k=c-\frac{b^2}{4a}$$

Find the conditions for the graph to cross the x-axis twice

A quadratic function crosses the x-axis twice if it has two distinct real roots. The discriminant, denoted by \(\Delta\), helps to determine the nature of the roots of a quadratic equation. The discriminant of the given quadratic function is: $$\Delta = b^2 - 4ac$$ For two distinct real roots, the discriminant must be greater than zero: $$\Delta > 0 \Rightarrow b^2 - 4ac > 0$$ Therefore, for the graph of the given quadratic function to cross the x-axis, the conditions on a, b, and c are: $$b^2 - 4ac > 0$$

Key concepts

Vertex of a quadratic

The vertex of a quadratic function provides valuable information about the function's graph, specifically the highest or lowest point of the parabola. For the quadratic function given by \( f(x) = ax^2 + bx + c \), the vertex can be found by rewriting the function in its vertex form, which is \( f(x) = a(x - h)^2 + k \). Here, \((h, k)\) represents the vertex.To identify \(h\) and \(k\):

• Complete the square to transform the standard form of the quadratic into the vertex form.
• The value of \(h\) can be found using the formula \( h = -\frac{b}{2a} \), indicating the x-coordinate of the vertex.
• The value of \(k\) is calculated as \( k = c - \frac{b^2}{4a} \), representing the y-coordinate of the vertex.

This process not only makes it easy to find the vertex coordinates but also aids in graphing the parabola and identifying its maximum or minimum point.

Discriminant

The discriminant is a powerful tool in understanding the nature of roots of a quadratic equation. For a quadratic function \( f(x) = ax^2 + bx + c \), the discriminant \( \Delta \) is derived from its standard quadratic formula and is defined as:\[ \Delta = b^2 - 4ac \]The value of the discriminant tells us about the roots of the quadratic:

• \( \Delta > 0 \): The quadratic has two distinct real roots, meaning the graph crosses the x-axis at two different points.
• \( \Delta = 0 \): The quadratic has exactly one real root or a repeated root, indicating the graph touches the x-axis once.
• \( \Delta < 0 \): The quadratic has no real roots, implying the graph does not cross the x-axis at all.

In this exercise, for the function to cross the x-axis twice, the discriminant must be greater than zero, implying the condition \( b^2 - 4ac > 0 \). This ensures two distinct real roots.

Completing the square

Completing the square is a method used to convert a quadratic function into its vertex form, which is crucial for identifying the vertex of the function. This process involves several calculated steps:1. Start with the standard quadratic form \( f(x) = ax^2 + bx + c \).2. Focus on the terms involving \( x \): \( ax^2 + bx \).3. Factor out \( a \) from these terms if \( a eq 1 \), to simplify: \[ a(x^2 + \frac{b}{a}x) \]4. Find the constant such that the inside becomes a perfect square trinomial: - Divide the coefficient of \( x \) (\( \frac{b}{a} \)) by 2 and square it, so it becomes \( \left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} \).5. Add and subtract this square inside the expression: \[ a(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} - \frac{b^2}{4a^2}) \]6. Group to form a complete square and adjust the constant term: \[ a(x + \frac{b}{2a})^2 - \frac{b^2}{4a} + c \]Finally, the quadratic is in vertex form, \( f(x) = a(x - h)^2 + k \), making it easy to identify the vertex at \((h, k)\) where \( h = -\frac{b}{2a} \) and \( k = c - \frac{b^2}{4a} \). This method not only finds the vertex but also helps visually understand the graph's structure.