Problem 14 - 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. $$ f(x)=x^{2}-10 x+18 $$

Step-by-Step Solution

Answer

The estimated roots are 3.1 and 6.9. The exact roots are \( x = 5 \pm \sqrt{7} \).

Step 1: Write the function in standard form

The given function is already in quadratic standard form, which is:\[ f(x) = x^2 - 10x + 18 \]

Step 2: Sketch the graph of the quadratic function

A quadratic function \( f(x) = ax^2 + bx + c \) is a parabola. Since the coefficient of \( x^2 \) (i.e., \( a = 1 \)) is positive, the parabola opens upwards. Find the vertex using the formula \( x = -\frac{b}{2a} \). Substituting \( a = 1 \) and \( b = -10 \), we get:\[ x = -\frac{-10}{2 \times 1} = 5 \]Substitute \( x = 5 \) into \( f(x) \) to get the vertex:\[ f(5) = 5^2 - 10(5) + 18 = -7 \]Thus, the vertex is \((5, -7)\). The y-intercept is \( c = 18 \). Plot the vertex, y-intercept, and sketch the parabola through these points and symmetrical around the line \( x = 5 \).

Step 3: Estimate the roots from the graph

Look at the points where the graph crosses the x-axis. The x-intercepts are the roots of the function. The graph suggests these points are around \( x = 3.1 \) and \( x = 6.9 \).

Step 4: Use the quadratic formula to find exact roots

The quadratic formula is given by: \\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]For the function \( f(x) = x^2 - 10x + 18 \), we have \( a = 1 \), \( b = -10 \), \( c = 18 \). Substitute these values:\[ x = \frac{10 \pm \sqrt{(-10)^2 - 4(1)(18)}}{2(1)} \]\[ x = \frac{10 \pm \sqrt{100 - 72}}{2} \]\[ x = \frac{10 \pm \sqrt{28}}{2} \]\[ x = \frac{10 \pm 2\sqrt{7}}{2} \]Simplify:\[ x = 5 \pm \sqrt{7} \] Thus, the exact roots are \( x = 5 + \sqrt{7} \) and \( x = 5 - \sqrt{7} \).

Key Concepts

Irrational Roots

Irrational roots are solutions to equations that cannot be expressed as exact fractions or integers, meaning they are not whole numbers or fractions. Instead, these roots often include square roots of numbers that are not perfect squares. For instance, in our worked example, the function \( f(x) = x^2 - 10x + 18 \) leads to the roots \( x = 5 + \sqrt{7} \) and \( x = 5 - \sqrt{7} \). Both involve the square root of 7, which is an irrational number because 7 is not a perfect square.

To simplify expressions involving irrational roots, it's helpful to recognize that only square roots of non-perfect squares are irrational. An irrational number can be represented as a decimal that goes on forever without repeating. However, we typically express them using radicals for precision and simplicity, such as \( 5 \pm \sqrt{7} \).

Understanding irrational roots is essential when solving quadratic equations. Even though they can seem complicated, these roots follow predictable patterns you can learn to recognize.

Parabola Graphing

Graphing a parabola involves visualizing the quadratic function. For example, any function of the form \( f(x) = ax^2 + bx + c \) results in a graph that is a curve called a parabola.

• This parabola's direction—whether it opens up or down—depends on the sign of \( a \). If \( a > 0 \), the parabola opens upward; if \( a < 0 \), it opens downward.
• The vertex, or the 'tip,' of this parabola provides significant insight into its shape and position. You calculate this point using the formula \( x = -\frac{b}{2a} \).
• For the function \( f(x) = x^2 - 10x + 18 \), \( a = 1 \), so it opens upwards. The vertex calculated at \( (5, -7) \) helps place the curve accurately on a graph.
• The symmetry in a parabola means if the vertex is centered at \( x = 5 \), then each point on the curve has a symmetrical counterpart across this vertical line, known as the axis of symmetry.

Graphing a parabola efficiently involves pinpointing the vertex, drawing the axis of symmetry, marking y-intercepts, and sketching the curve through these reference points. This analytic understanding helps predict the shape and position of any parabola quickly.

Quadratic Formula

The quadratic formula is a universal solution method for any quadratic equation of the form \( ax^2 + bx + c = 0 \). This essential tool allows you to find roots directly, using the coefficients \( a \), \( b \), and \( c \). The formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula derives from completing the square of the quadratic equation and provides both real and complex roots, depending on the discriminant \( b^2 - 4ac \).

• If \( b^2 - 4ac > 0 \), we have two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root, also called a repeated root.
• If \( b^2 - 4ac < 0 \), the roots are complex and appear in conjugate pairs.

For the function \( f(x) = x^2 - 10x + 18 \), using the quadratic formula leads to the roots \( 5 \pm \sqrt{7} \), as shown in the solution. These roots are found by identifying the coefficients for \( a = 1 \), \( b = -10 \), and \( c = 18 \), then substituting them into the formula.

Using the quadratic formula replaces guesswork with certainty and precision, making it invaluable not just in solving equations but also in understanding the nature of their solutions.