Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-22: (a) Use the quadratic formula to find the zeros of \(f\). (b) Find the maximum or minimum value of \(f(x)\). (c) Sketch the graph of \(f\). $$ f(x)=x^{2}-4 x $$

Step-by-Step Solution

Answer

Zeros at \(x = 0\) and \(x = 4\); minimum value of \(f(x)\) is \(-4\) at \(x = 2\).

Step 1: Identify the coefficients

For a quadratic equation in the standard form \(ax^2 + bx + c\), identify the coefficients: \(a = 1\), \(b = -4\), and \(c = 0\) from the given \(f(x) = x^2 - 4x\).

Step 2: Apply the quadratic formula

The quadratic formula is \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\). Substitute the identified coefficients into the formula: \[ x = \frac{{-(-4) \pm \sqrt{{(-4)^2 - 4 \cdot 1 \cdot 0}}}}{2 \cdot 1} \]Simplify the expression: \[ x = \frac{{4 \pm \sqrt{16}}}{2} \]This further simplifies to \[ x = \frac{{4 \pm 4}}{2} \]

Step 3: Solve for the zeros

Calculate the two possible values for \(x\) from the expression in Step 2:\[ x = \frac{{4 + 4}}{2} = 4 \quad \text{and} \quad x = \frac{{4 - 4}}{2} = 0 \]Thus, the zeros of \(f(x)\) are \(x = 4\) and \(x = 0\).

Step 4: Determine the vertex of the parabola

The vertex of a parabola given by \(y = ax^2 + bx + c\) occurs at \(x = -\frac{b}{2a}\). Substitute \(b = -4\) and \(a = 1\) into the formula:\[ x = -\frac{-4}{2 \cdot 1} = 2 \]Then, substitute \(x = 2\) into \(f(x)\) to find the \(y\)-value:\[ f(2) = 2^2 - 4 \times 2 = 4 - 8 = -4 \]Thus, the vertex is \((2, -4)\), indicating a minimum point.

Step 5: Plot and sketch the graph

To sketch the graph, note the following:- The zeros (x-intercepts) are at \(x = 0\) and \(x = 4\).- The vertex is at \((2, -4)\) and is a minimum point.- Since the coefficient of \(x^2\) is positive, the parabola opens upward.- Plot these key points and draw a smooth curve through them to complete the sketch.

Key Concepts

Parabola

A parabola is a U-shaped curve that represents the graph of a quadratic function. Quadratic functions have the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

• The direction the parabola opens (upward or downward) is determined by the sign of the coefficient \(a\).
• If \(a\) is positive, the parabola opens upwards, forming a 'U' shape.
• If \(a\) is negative, the parabola opens downwards, forming an inverted 'U'.

For example, in the function \(f(x) = x^2 - 4x\), the coefficient of \(x^2\) is 1, which is positive. Therefore, this parabola opens upwards. Understanding how the coefficient of \(x^2\) changes the graph helps in anticipating the general shape of the graph, which is crucial when sketching quadratics.

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations. A quadratic equation is typically written as \(ax^2 + bx + c = 0\). The quadratic formula is given by:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]To use this formula, it's important to identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation.
In our example \(f(x) = x^2 - 4x\) rewritten as \(x^2 - 4x + 0 = 0\), we found \(a = 1\), \(b = -4\), and \(c = 0\). These coefficients are substituted into the quadratic formula to find the values of \(x\) that make the function zero.
The formula involves:

• Calculating the discriminant \(b^2 - 4ac\), which determines the nature of the roots (real or complex).
• Using the '+' and '-' symbols in the formula to find two possible solutions for \(x\).

It serves as a universal method for solving any quadratic equation, making it a powerful and versatile mathematical tool.

Zeros of a Function

The zeros of a function are the points where the graph crosses the x-axis. At these points, the value of the function is zero. Finding the zeros of a quadratic function is crucial as they show where the parabola intersects the x-axis.
To find the zeros, use the quadratic formula to solve \(ax^2 + bx + c = 0\). In our example, after substituting in the values, we simplified:\[x = \frac{{4 \pm 4}}{2}\]This resulted in two zeros: \(x = 4\) and \(x = 0\).

• The zeros can be considered as solutions of the quadratic equation.
• They are also called roots or x-intercepts.

Knowing the zeros helps when sketching the graph, as these are key points where the graph will touch or cross the x-axis.

Vertex of a Parabola

The vertex of a parabola represents the highest or lowest point on the graph. It occurs at the axis of symmetry, a vertical line that passes through the vertex. For a parabola given by the equation \(y = ax^2 + bx + c\), the x-coordinate of the vertex can be found using:\[x = -\frac{b}{2a}\]
In our quadratic function \(f(x) = x^2 - 4x\), we calculated:\[x = -\frac{-4}{2 \cdot 1} = 2\]By substituting \(x = 2\) back into the function, we found the y-coordinate to be \(-4\), making the vertex \((2, -4)\).
The vertex provides important information about the parabola:

• If the parabola opens upwards, the vertex represents the minimum value of the function.
• If the parabola opens downwards, the vertex represents the maximum value.
• The vertex is a critical point for analyzing the function's properties, such as its range and domain.

Thus, the vertex is a pivotal element in sketching and understanding the graphical representation of quadratic functions.