Problem 12 - Algebra 2 and Trigonometry

Question

In \(9-14 :\) a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form. $$ f(x)=x^{2}-6 x+6 $$

Step-by-Step Solution

Answer

Roots are approximately 1.6, 4.4; exact roots are \(x = 3 \pm \sqrt{3}\).

Step 1: Understand the Function

The given function is a quadratic function of the form \(f(x) = ax^2 + bx + c\), where \(a = 1\), \(b = -6\), and \(c = 6\). This function represents a parabola.

Step 2: Sketch the Graph of the Function

To sketch the graph, we identify the vertex and axis of symmetry. The vertex form of a quadratic function is given by \( f(x) = a(x-h)^2 + k \). Using the formula for the vertex, \(h = -\frac{b}{2a}\), we find \(h = 3\). Plug \(h\) back into the function to find \(k\), resulting in \(k = f(3) = 3^2 - 6  3 + 6 = -3\). The vertex is (3, -3). The axis of symmetry is \(x = 3\). Now plot the vertex and a few additional points, like \(f(0) = 6\) and mirror them across the axis.

Step 3: Estimate the Roots from the Graph

The graph intersects the x-axis at the roots of the function. From the sketch, estimate these points visually. They appear at approximately \(x  1.6\) and \(x  4.4\).

Step 4: Calculate Exact Roots Using the Quadratic Formula

Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 1\), \(b = -6\), and \(c = 6\):\[x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4  1  6}}{2  1}\]\[x = \frac{6 \pm \sqrt{36 - 24}}{2}\]\[x = \frac{6 \pm \sqrt{12}}{2}\]Simplify \(\sqrt{12} = 2\sqrt{3}\):\[x = \frac{6 \pm 2\sqrt{3}}{2}\]\[x = 3 \pm \sqrt{3}\]

Step 5: Express the Exact Irrational Roots

From the previous calculation, the exact irrational roots are \(x = 3 + \sqrt{3}\) and \(x = 3 - \sqrt{3}\).

Key Concepts

Parabola

A quadratic function like \(f(x) = x^2 - 6x + 6\) forms a curve known as a parabola when graphed. Parabolas are symmetrical U-shaped graphs that can open upwards or downwards depending on the leading coefficient (in this case, it is 1, so it opens upwards).
Parabolas have key features: a vertex, an axis of symmetry, and roots. The vertex is the highest or lowest point on the graph, depending on how the parabola opens. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. Roots are the x-values where the parabola passes through the x-axis.

• **Upward Openness:** Since the coefficient of \(x^2\) is positive, the parabola opens upwards.
• **Vertex:** This is the turning point of the parabola located at (3, -3), indicating the minimum point on the graph.
• **Axis of Symmetry:** This vertical line is \(x = 3\). Every point on the left of this line has a corresponding point on the right, mirroring across this axis.

Understanding these features helps in visualizing and sketching the parabolic graph accurately.

Vertex Form

The vertex form of a quadratic function is a way to express the function that clearly shows the vertex, making graphing much easier.
This form is expressed as: \[ f(x) = a(x-h)^2 + k \]Here, \(a\) affects the "width" and "direction" (up or down) of the parabola, while \((h, k)\) are the coordinates of the vertex. For the function \(f(x) = x^2 - 6x + 6\), we first find \(h\) using the formula \(h = -\frac{b}{2a}\).

• **Finding \(h\):** Substitute \(a = 1\) and \(b = -6\) into the formula: \[ h = -\frac{-6}{2 \times 1} = 3 \]
• **Finding \(k\):** Plug \(h\) into the function: \[ k = f(3) = 3^2 - 6 \times 3 + 6 = -3 \]

The vertex form thus reveals that the vertex of this parabola is at (3, -3). This makes it straightforward to plot the parabola on a graph.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots (solutions) of quadratic equations. For the equation \(ax^2 + bx + c = 0\), the roots can be found using: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula considers both the coefficient values and calculates the discriminant, \(b^2 - 4ac\), under the square root.

• **Using the Formula:** For \(f(x) = x^2 - 6x + 6\), we identify \(a = 1\), \(b = -6\), and \(c = 6\). By substituting these into the quadratic formula: \[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \times 1 \times 6}}{2 \times 1} \] \[ x = \frac{6 \pm \sqrt{36 - 24}}{2} \] \[ x = \frac{6 \pm \sqrt{12}}{2} \]
• **Resulting Roots:** This simplifies further to: \[ x = 3 \pm \sqrt{3} \]

The plus-minus symbol (±) indicates two possible roots, corresponding to the points where the parabola crosses the x-axis.

Exact Irrational Roots

Exact irrational roots refer to the precise roots of a quadratic equation, which are often expressed in terms of square roots rather than decimals. They are especially useful in math because they offer an exact result without approximation.
For our function \(f(x) = x^2 - 6x + 6\), the roots are: \[ x = 3 + \sqrt{3} \] \[ x = 3 - \sqrt{3} \] Unlike approximate decimal roots, these roots remain unrounded and exhibit their true mathematical values. This retention of precision is vital in advanced calculations and higher-level mathematics

• **Meaning of Exactness:** "Exact" means the computation involves no rounding, preserving the purity of the mathematical expression.
• **Importance in Mathematics:** Exact roots, especially the irrational ones, provide deeper insights into the nature of the equations and functions they are solutions for.

These roots give us the exact points where the parabola intersects the x-axis, useful in thorough mathematical analysis.