Problem 10 - 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. $$ \mathrm{f}(x)=x^{2}-2 x-2 $$

Step-by-Step Solution

Answer

The approximate roots are -0.7 and 2.7; the exact roots are \(1 + \sqrt{3}\) and \(1 - \sqrt{3}\).

Step 1: Identify the Type of Function

The given function \( f(x) = x^2 - 2x - 2 \) is a quadratic function. This type of function generally has the form \( ax^2 + bx + c \) and graphs as a parabola.

Step 2: Determine the Vertex of the Parabola

The vertex of a parabola in the form \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \), \( b = -2 \). Plug in the values: \( x = -\frac{-2}{2 \times 1} = 1 \). Substitute \( x = 1 \) back into the function to find \( y \): \( f(1) = 1^2 - 2 \times 1 - 2 = -3 \). Therefore, the vertex is \( (1, -3) \).

Step 3: Sketch the Parabola

Draw a vertical axis for \( y \) and a horizontal axis for \( x \). Plot the vertex at \( (1, -3) \). Since the parabola opens upwards (\( a > 0 \)), draw the parabola with the vertex as the lowest point. Use additional points like \( f(0) = -2 \) and \( f(2) = -2 \) to guide the shape of the graph.

Step 4: Estimate the Roots from the Graph

The roots correspond to the x-intercepts of the function. Observe where the parabola crosses the x-axis. By estimation, it appears the roots are around \( x = -0.7 \) and \( x = 2.7 \), which are the approximate points where the graph crosses the x-axis.

Step 5: Solve for Exact Roots Using the Quadratic Formula

The roots of a quadratic equation \( ax^2 + bx + c = 0 \) can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Insert \( a = 1 \), \( b = -2 \), and \( c = -2 \): \[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \times 1 \times (-2)}}{2 \times 1} = \frac{2 \pm \sqrt{4 + 8}}{2} = \frac{2 \pm \sqrt{12}}{2} = \frac{2 \pm 2\sqrt{3}}{2} = 1 \pm \sqrt{3} \]. So the exact roots are \( 1 + \sqrt{3} \) and \( 1 - \sqrt{3} \).

Key Concepts

Parabola

A parabola is the graph of a quadratic function and it has a unique, symmetrical U-shape. When you see a function like \( f(x) = x^2 - 2x - 2 \), you're dealing with a parabola. The general equation for a quadratic function is \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.

• If \( a > 0 \), the parabola opens upwards, resembling a smile.
• If \( a < 0 \), it opens downwards like a frown.

In our case, since \( a = 1 \), the parabola opens upwards. The symmetry of a parabola is centered around its vertex, which we'll explore more in the next section. Drawing parabolas requires identifying a few key points: the vertex, some additional points, and the shape (either opening up or down). Be sure to plot these carefully to achieve a true representation.

Quadratic Formula

The quadratic formula is a mathematical tool used to find the roots or solutions of quadratic equations of the form \( ax^2 + bx + c = 0 \). This formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows you to solve any quadratic equation, even when factorization is difficult or impossible. In the exercise, substituting \( a = 1 \), \( b = -2 \), and \( c = -2 \) into the formula gave us the roots: \( 1 + \sqrt{3} \) and \( 1 - \sqrt{3} \).

• The term under the square root, \( b^2 - 4ac \), is called the discriminant.
• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is one real root (a double root).
• If negative, there are no real roots, but two complex roots.

The quadratic formula can always be used to find exact roots.

Vertex

The vertex of a parabola is a crucial point where the curve changes direction. For the quadratic function in standard form \( ax^2 + bx + c \), the vertex can be found using the formula:\[ x = -\frac{b}{2a}\]In our problem, using \( a = 1 \) and \( b = -2 \), we determined that the x-coordinate of the vertex is \( 1 \). By substituting this back into the function, we calculated the y-coordinate as \( -3 \). Therefore, the vertex of this parabola is at \( (1, -3) \).
The vertex tells us the minimum or maximum point of the parabola:

• If the parabola opens upwards (\( a > 0 \)), the vertex is the minimum point.
• Conversely, if it opens downwards (\( a < 0 \)), the vertex represents the maximum point.

The vertex form of a quadratic equation, \( a(x-h)^2 + k \), directly reveals the vertex \((h, k)\). Understanding the vertex assists in graphing and interpreting the function's behavior.

Roots

Roots of a quadratic function are the values of \( x \) that make the function equal zero, representing where the parabola crosses the x-axis. These are often referred to as solutions or zeros of the function. In the given equation, these roots were calculated using both graph estimation and the quadratic formula.
Stages to find roots:

• Initial estimation from the graph gave approximate roots \( x \approx -0.7 \) and \( x \approx 2.7 \).
• Exact values were found using the quadratic formula as \( 1 + \sqrt{3} \) and \( 1 - \sqrt{3} \).

Finding roots is essential because it tells us where the graph hits the x-axis. Knowing these points can provide insight into the function's real-world applications, like determining when an object hits the ground if it follows a parabolic path. Always remember to check your work by plugging these values back into the original equation to ensure they make the function zero. This step is an excellent way to verify the correctness of your results.