Problem 11 - 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. $$ f(x)=x^{2}+4 x+2 $$

Step-by-Step Solution

Answer

The estimated roots are about -3.4 and -0.6; the exact roots are \(-2 \pm \sqrt{2}\).

Step 1: Write the Equation in Function Form

The given function is already in the quadratic form: \(f(x) = x^2 + 4x + 2\). This is a parabola that can be sketched on a graph to find its roots.

Step 2: Sketch the Graph of the Quadratic Function

To sketch the graph of \(f(x) = x^2 + 4x + 2\), identify the key features. The graph is a parabola that opens upwards because the coefficient of \(x^2\) is positive. The vertex can be found using \(x = -\frac{b}{2a}\), where \(a = 1\) and \(b = 4\). Calculating gives \(x = -2\). Substitute \(x = -2\) into the function to find \(y = -2\). Thus, the vertex is \((-2, -2)\). Draw this parabola on the graph.

Step 3: Estimate the Roots from the Graph

From the graph, observe where the parabola intersects the x-axis. These x-values are the roots of the function. Estimate these roots to the nearest tenth. The graph might show intersections near \(x = -3.4\) and \(x = -0.6\).

Step 4: Find the Exact Irregular Roots Using the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the exact roots. Here, \(a = 1\), \(b = 4\), \(c = 2\). Calculate the discriminant: \(b^2 - 4ac = 16 - 8 = 8\). Substitute into the formula to get \(x = \frac{-4 \pm \sqrt{8}}{2}\). Simplify \(\sqrt{8}\) to \(2\sqrt{2}\), resulting in the roots \(x = -2 \pm \sqrt{2}\). These are the exact irrational roots.

Key Concepts

Graphing Parabolas

Graphing a parabola means drawing the u-shaped curve of a quadratic function on a coordinate plane. This curve can help us find important information about the quadratic function, like its roots or maximum and minimum points. For the function \( f(x) = x^2 + 4x + 2 \), we are dealing with a standard form quadratic function, which is expressed generally as \( ax^2 + bx + c \).
The coefficient of \( x^2 \) in this equation is positive, which means the parabola will open upwards.

• The vertex of a parabola is a special point where it changes direction and can be found using the formula \( x = -\frac{b}{2a} \).
• Substitute this \( x \) value back into the equation to find the corresponding \( y \)-value and get the coordinates of the vertex.
• For the given function, the vertex is at \((-2, -2)\), meaning that the lowest point of the graph is at \( y = -2 \).

Place the vertex on a graph, then use additional points or a general shape understanding to draw the parabola. This visualization can guide us to estimate where the parabola crosses the x-axis, indicating potential roots.

Estimating Roots

Roots, also known as zeros or x-intercepts, are where the graph of the parabola crosses the x-axis. These are the values of \( x \) that solve the equation \( f(x) = 0 \). When graphing, it's essential to look at the points where the parabola hits the x-axis.

• For rough estimates, look at the graph and determine the x-values where the curve meets the axis.
• In our example, the estimated roots found by looking at the graph are around \(-3.4\) and \(-0.6\).

Estimating roots by hand from a graph is a great way to get a quick idea, but these estimates might not be exact. The precision can vary depending on how accurately the graph is drawn or how detailed it is.

Quadratic Formula

The quadratic formula is an important tool for finding the exact roots of a quadratic equation. This formula is particularly useful when the roots of the equation aren't perfect integers.

• The formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a \), \( b \), and \( c \) are constants from the quadratic equation \( ax^2 + bx + c = 0 \).
• Start by calculating the discriminant, \( b^2 - 4ac \).
• Substitute the values into the formula to find the roots.

In our specific function \( f(x) = x^2 + 4x + 2 \), after calculating the discriminant and using the formula, the roots are \( x = -2 \pm \sqrt{2} \). These roots are exact solutions to the equation set to zero.

Irrational Roots

Irrational roots are values that cannot be expressed as simple fractions. They often result from the quadratic formula when the discriminant is not a perfect square. In this case, our discriminant \( 8 \) results in an irrational number when we take its square root.

• The expression \( \sqrt{8} \) simplifies to \( 2\sqrt{2} \), revealing the roots as \( x = -2 \pm \sqrt{2} \).
• Irrational roots can be represented in simplest radical form to provide the most precise solution.

These roots may not be rational numbers but are still precise exact solutions that define the intersection points of the parabola with the x-axis. Using this approach, not only do we find accurate roots, but we also gain insight into the nature of the solutions involved. Recognizing when roots are irrational helps to understand and communicate the behavior of quadratic functions.