Problem 26 - Introductory Algebra for College Students

Question

Graph the parabola whose equation is given $$y=x^{2}+10 x+9$$

Step-by-Step Solution

Answer

The graph of the given function \(y=x^{2}+10 x+9\) is a parabola with vertex \((-5, -16)\), and axis of symmetry \(x=-5\).

Step 1: Identify the form of the parabola

The given parabola \(y=x^{2}+10x+9\) is in the form \(y=ax^{2}+bx+c\) where \(a=1\), \(b=10\), and \(c=9\).

Step 2: Find the vertex

The vertex of a parabola \(y = ax^2 + bx + c\) is given by \((-\frac{b}{2a}, f(-\frac{b}{2a}))\). Substituting \(a=1\) and \(b=10\) into \(-\frac{b}{2a}\), we get \(-\frac{10}{2*1} = -5\). Substituting \(x=-5\) into \(y=x^{2}+10x+9\), we get \(y=(-5)^{2}+10*(-5)+9 = -16\). Thus, the vertex of the parabola is \((-5,-16)\).

Step 3: Identify the axis of symmetry

The axis of symmetry of a parabola \(y = ax^2 + bx + c\) is the vertical line \(x = -\frac{b}{2a}\). So for this parabola, the axis of symmetry is \(x=-5\).

Step 4: Sketch the parabola

Now we have all the information we need to sketch the parabola. Draw an x-y coordinate plane, and mark the vertex point \((-5,-16)\) and plot the axis of symmetry \(x=-5\). Since \(a=1 > 0\), the parabola opens upwards. Sketch the parabola so that it is symmetric about the line \(x=-5\), and passes through the vertex.

Key Concepts

Vertex Form

The vertex form of a quadratic equation is one of the most informative and visually appealing ways to express a parabola. It is represented as \[ y = a(x-h)^2 + k \] where

• \( (h, k) \) is the vertex of the parabola,
• \(a\) indicates the direction and the width of the parabola.

This form is particularly useful because it readily provides the vertex, a crucial feature when graphing. The vertex \( (h, k) \) is the highest or lowest point of the parabola, depending on whether the parabola opens downward or upward. An upward opening is characterized by \( a > 0 \), and a downward opening when \( a < 0 \).

The conversion from standard form \( y = ax^2 + bx + c \) to vertex form involves completing the square, which can sometimes be complex for beginners. That's why having a grasp on what the vertex form describes is essential for transforming and graphing quadratic equations effectively. Understanding the vertex form means you can easily identify the vertex on a graph, making it easier to sketch and visualize the parabola.

Axis of Symmetry

The axis of symmetry is an important concept when visualizing or graphing a parabola. It is the vertical line that acts as a mirror, reflecting every point on one side of the parabola onto the other. For any parabola given in standard form \( y = ax^2 + bx + c \), the axis of symmetry is found with the formula\[ x = -\frac{b}{2a} \] This line passes through the vertex, meaning it divides the parabola into two symmetrical halves.

In the case of a quadratic equation like \( y = x^2 + 10x + 9 \), we determined earlier that \(-\frac{b}{2a} = -5\). Thus, the parabola is symmetric about the vertical line \( x = -5 \). This property is useful for both graphing and understanding the behavior of parabolas since it provides insights into the arrangement of points as well as vertex position. Knowing the axis of symmetry lets you plot more points accurately and ensures your graph is as precise as possible.

Quadratic Equation

A quadratic equation is any equation that can be rearranged into the form \( ax^2 + bx + c = 0 \). This type of equation describes a parabola when graphed on a coordinate plane, appearing as a U-shaped curve. What makes quadratics interesting is their wide application, from physics to finance.

• \( a\) determines the direction and steepness of the parabola,
• \( b\) influences the parabola's vertex's x-coordinate,
• \( c\) impacts the y-intercept of the parabola.

Standard form, as shown, is typically the starting point for solving or graphing quadratics. By identifying the coefficients \(a\), \(b\), and \(c\), you can find key features like the vertex and axis of symmetry.

Solving quadratic equations can also involve factoring, using the quadratic formula, or completing the square. Each method has its own advantages and can depend on the specific form of the equation or the values of the coefficients. Understanding these basics ensures a strong foundation in mathematics and enables you to tackle a variety of analytical challenges effectively.