Problem 12 - Algebra 2

Question

Graph each function. Label the vertex and the axis of symmetry. $$ y=x^{2}+4 x+1 $$

Step-by-Step Solution

Answer

The vertex is at (-2, -3) and the axis of symmetry is the line x=-2. After plotting the vertex and additional points, graph the upward-opening parabola.

Step 1: Determine the Coordinates of the Vertex

To find the vertex of a parabola in the form of y=ax^2+bx+c, use the vertex formula x=-b/(2a). In this equation, a=1 and b=4, so the vertex x-coordinate is x=-4/(2*1)=-2. Plug x=-2 into the equation to find the y-coordinate: y=(-2)^2+4*(-2)+1=4-8+1=-3. So the vertex is at (-2, -3).

Step 2: Identify the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex. Since the x-coordinate of the vertex is -2, the equation for the axis of symmetry is x=-2.

Step 3: Plot the Vertex and the Axis of Symmetry

Plot the vertex point (-2, -3) on the graph. Draw a dashed vertical line through x=-2 to represent the axis of symmetry.

Step 4: Find Additional Points

Choose x-values to the left and right of the vertex to find additional points to plot. For example, if x=-3 or x=-1, plug these x-values into the equation to find the corresponding y-values.

Step 5: Sketch the Parabola

Using the symmetrical points from Step 4, draw a smooth curve through these points and the vertex to form the parabola. Make sure the parabola opens upwards since the coefficient of x^2 is positive.

Step 6: Label the Vertex and Axis of Symmetry

Clearly mark the vertex on the graph and label it with its coordinates (-2, -3). Also, label the axis of symmetry with its equation, x=-2.

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial for graphing quadratic functions. The vertex is the highest or lowest point on the parabola and serves as a pivotal element of its shape. For the function \(y = x^2 + 4x + 1\), we calculate the vertex by using the formula \(-\frac{b}{2a}\).

Considering our equation, \(a = 1\) and \(b = 4\), we find the x-coordinate of the vertex to be \(x = -\frac{4}{2\cdot1} = -2\). Substituting \(x = -2\) back into the function gives us the y-coordinate: \((-2)^2 + 4(-2) + 1 = -3\). Therefore, the vertex of the parabola for this function is at the coordinates \( (-2, -3) \). It is essential to graph the vertex accurately, as it will guide the parabola's shape and position on the coordinate plane.

Axis of Symmetry

The axis of symmetry in a parabolic graph is a vertical line that divides the parabola into two mirror images. It goes directly through the vertex, reflecting the symmetry of the parabola's halves. For our example \(y = x^2 + 4x + 1\), the axis of symmetry can be found using the vertex's x-coordinate.

Since the x-coordinate of the vertex is \(x = -2\), the axis of symmetry's equation is also \(\text{x = -2}\). When you graph the parabola, this line helps ensure that the shape is symmetrical. Draw a dashed line on the graph to represent the axis of symmetry, ensuring it passes through the vertex point \( (-2, -3) \). This line is not only a guide for graph symmetry but also an integral feature that characterizes the parabola's properties.

Quadratic Functions

Quadratic functions form the backbone of parabolas and are easily recognizable by their standard form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). These functions are known for their curved graphs, which are called parabolas, and can open upwards or downwards based on the sign of the coefficient \(a\).

For the function in our exercise \(y = x^2 + 4x + 1\), the positive coefficient of \(x^2\) indicates that the parabola opens upwards. This is confirmed during graphing, as we plot additional points around the vertex to see the shape of the parabola. The symmetry about the axis, the vertex's position, and the direction in which the parabola opens are all determined by the function's coefficients. An understanding of these characteristics is key to successfully graphing the curve and comprehending the behavior of quadratic functions.