Problem 19 - 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 \(f\). $$ f(x)=x^{2}+4 x+9 $$

Step-by-Step Solution

Answer

The function has no real zeros, has a minimum value of 5 at \( x = -2 \), and is a parabola opening upwards.

Step 1: Identify the Coefficients

The quadratic function given is \( f(x) = x^2 + 4x + 9 \). The coefficients are \( a = 1 \), \( b = 4 \), and \( c = 9 \).

Step 2: Apply the Quadratic Formula

We use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute \( a = 1 \), \( b = 4 \), and \( c = 9 \) into the formula to find the zeros of the function.

Step 3: Calculate the Discriminant

The discriminant \( b^2 - 4ac \) is calculated as follows: \( 4^2 - 4 \times 1 \times 9 = 16 - 36 = -20 \). The discriminant is negative, which indicates that the quadratic equation has no real zeros.

Step 4: Find the Vertex using Vertex Formula

The vertex of the parabola can be found using \( x = -\frac{b}{2a} \). Substitute \( a = 1 \) and \( b = 4 \) into the formula: \( x = -\frac{4}{2 \times 1} = -2 \).

Step 5: Calculate the Function Value at the Vertex

Substitute \( x = -2 \) back into the function to find the minimum value: \( f(-2) = (-2)^2 + 4(-2) + 9 = 4 - 8 + 9 = 5 \). Therefore, the minimum value of the function is \( 5 \) at \( x = -2 \).

Step 6: Sketch the Graph

Since the parabola opens upwards (as \( a = 1 > 0 \)), the vertex \((-2, 5)\) is a minimum point. With no real zeros, the graph does not cross the x-axis. The parabola is symmetric about the line \( x = -2 \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It allows us to find the zeros, or roots, of any quadratic function. The formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula is derived from completing the square on the general quadratic equation and is applicable for any quadratic equation.
To use the quadratic formula, one must identify the coefficients \(a\), \(b\), and \(c\) from the equation. By substituting these values into the formula, you can calculate the roots of the equation. However, note that when the discriminant \(b^2 - 4ac\) is negative, as seen in the original exercise, no real zeros exist, which means the graph of the quadratic does not intersect the x-axis.

Vertex of a Parabola

The vertex of a parabola is a key feature that represents the highest or lowest point on the graph, depending on whether it opens upwards or downwards. Finding the vertex is simple when using the vertex formula, which is:
\[x = -\frac{b}{2a}\]The x-value of the vertex can be found by substituting the coefficients \(a\) and \(b\) into this formula. Once the x-coordinate is determined, substitute it back into the quadratic function to find the corresponding y-coordinate. This gives the vertex point \((x, f(x))\).
In the problem, for \(f(x) = x^2 + 4x + 9\), the vertex is found to be at \(x = -2\). Substituting \(-2\) back into the function gives you the minimum value of the parabola, \((x, y) = (-2, 5)\). The vertex can also hint whether the parabola's minimum or maximum; when \(a > 0\), the parabola opens upwards, thus having a minimum at the vertex.

Discriminant in Quadratic Equations

The discriminant is a crucial part of the quadratic formula and plays a significant role in determining the nature of the roots of a quadratic equation. The discriminant is the expression under the square root in the quadratic formula:
\[b^2 - 4ac\]This value tells us whether the roots are real or complex:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant equals zero, there is exactly one real root (i.e., a repeated root).
• If the discriminant is negative, as with the exercise \(16 - 36 = -20\), no real roots exist; the roots are complex or imaginary numbers.

Understanding the discriminant helps us know not only about the roots but also the parabolic graph's interaction with the x-axis, crucial in sketching or analyzing the graph.

Graphing Parabolas

Graphing a parabola involves understanding its shape and features such as vertex, direction, and intercepts. A quadratic function, \(ax^2 + bx + c\), can be graphed as a curve called a parabola.
Key aspects to consider are:

• The parabola's direction: Determined by the coefficient \(a\). If \(a > 0\), it opens upward; if \(a < 0\), downward.
• The vertex: Known as the highest or lowest point, found using the vertex formula.
• Intercepts: Points where the parabola crosses the axes. The y-intercept is \(c\), the constant term in the equation.
• Axis of Symmetry: A vertical line that passes through the vertex, given by \(x = -\frac{b}{2a}\).

In the exercise, the parabola opens upward since \(a = 1\), has its vertex at \((-2, 5)\), and does not cross the x-axis due to a negative discriminant. As a result, the graph shows a curve rising from the vertex.