Problem 14 - Algebra 2

Question

Graph each function. Label the vertex and the axis of symmetry. $$ y=x^{2}+6 x+9 $$

Step-by-Step Solution

Answer

Vertex: \((-3,0)\), Axis of Symmetry: \(x = -3\). The parabola opens upwards, with the vertex at the lowest point.

Step 1: Identify the Quadratic Coefficients

The given quadratic function is in the standard form, where the coefficients from the equation \(y = ax^2 + bx + c\) can be identified as: \(a = 1\), \(b = 6\), and \(c = 9\).

Step 2: Find the Vertex

The vertex of a parabola given in the standard form \(y = ax^2 + bx + c\) can be found using the formula \(h = -\frac{b}{2a}\) for the x-coordinate, and then calculating \(k = f(h)\) for the y-coordinate of the vertex. For the given function, the vertex is at \(h = -\frac{6}{2 \cdot 1} = -3\) and \(k = (1)(-3)^2 + (6)(-3) + 9 = 0\), resulting in the vertex \((-3, 0)\).

Step 3: Determine the Axis of Symmetry

The axis of symmetry for a parabola is a vertical line that goes through the vertex. The equation for the axis of symmetry can be derived from the vertex's x-coordinate, given as \(x = h\). Thus, for our function, the axis of symmetry would be the line \(x = -3\).

Step 4: Plot the Vertex and Axis of Symmetry

Plot the vertex on the graph at the point \((-3, 0)\). Then draw the axis of symmetry: a vertical line through the vertex, at \(x = -3\).

Step 5: Sketch the Parabola

Since \(a = 1\) is positive, the parabola opens upwards. Use the symmetry of the graph about the axis of symmetry to plot additional points, mirroring points across the axis. Then, draw the parabola, making sure it is symmetric and passes through the vertex and the plotted points.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial concept in graphing quadratic functions. It is the highest or lowest point on the graph, representing a maximum or minimum when the parabola opens down or up, respectively. To find the vertex, you can use the formula:

Finding the Vertex

For a quadratic function in standard form, which is \(y = ax^2 + bx + c\), the x-coordinate of the vertex (h) can be calculated with \(h = -\frac{b}{2a}\). Once you have h, plug it into the original equation to find the y-coordinate (k), which gives the vertex as the ordered pair (h, k).
In the example problem, we calculate the vertex for \(y = x^2 + 6x + 9\) using the given coefficients. The steps provide a vertex at \((-3, 0)\), indicating this is the lowest point of the parabola since the leading coefficient, a, is positive. Understanding the vertex helps us visualize the parabola's shape and direction.

Axis of Symmetry

Closely tied to the concept of the vertex is the axis of symmetry. This imaginary line runs vertically through the vertex and serves as a mirror, reflecting one half of the parabola onto the other. The axis of symmetry can be simply defined by the equation \(x = h\), where h is the x-coordinate of the vertex.

Identifying the Axis

Identifying the axis of symmetry is a fundamental step in graphing parabolas as it ensures symmetry in the graph. For instance, in our example with the quadratic function \(y = x^2 + 6x + 9\), the axis of symmetry is found to be \(x = -3\) after determining the vertex. Knowing this, we’re able to sketch our graph with assurance that all points will reflect accurately across this line.

Quadratic Coefficients

The quadratic coefficients, denoted as a, b, and c, come from the quadratic function's standard form \(y = ax^2 + bx + c\). These coefficients play essential roles in shaping the graph of the parabola.

• a (leading coefficient): Determines whether the parabola opens upward (a>0) or downward (a<0) and affects the width of the parabola. The greater the absolute value of a, the narrower the parabola.
• b: Influences the position of the vertex along the x-axis.
• c: Represents the y-intercept, where the graph crosses the y-axis.

In the step-by-step exercise provided, the coefficients are identified as \(a = 1\), \(b = 6\), and \(c = 9\). These values are instrumental in determining the vertex, as mentioned earlier, and help in plotting the graph. Understanding each coefficient's impact can significantly improve the ease of graphing any quadratic function.