Question

Graph the equation. $$y=x^{2}+x+2$$

Short answer

The graph of the function \(y = x^{2} + x + 2\) is a parabola which opens upward. Its vertex is at \((-1/2, 7/4)\) and it has an axis of symmetry at \(x = -1/2\).

Step-by-step solution

Identify the Function Type

The function \(y = x^{2} + x + 2\) is a quadratic function as it fits the form \(ax^{2} + bx + c\), where a, b, and c are constants and a is not equal to zero.

Find the Vertex

The vertex can be found using the formula \(-b/2a\). In this case, a equals 1 and b equals 1. Therefore, the x-coordinate of the vertex, \(h\), would be \(-1/(2*1) = -1/2\). Substituting -1/2 into the equation for x gives the y-coordinate of the vertex, \(k\), as \(y = (-1/2)^{2} -1/2 + 2 = 7/4\). Thus, the vertex of the graph is \((-1/2, 7/4)\).

Determine the Opening

For a quadratic function in the form of \(y = ax^{2} + bx + c\), if \(a > 0\), the graph opens upward; if \(a < 0\), the graph opens downward. In this case, since a is 1 and 1 is greater than zero, the graph will open upward.

Identify the Axis of Symmetry

The axis of symmetry is a vertical line running through the vertex of the parabola. It has the equation \(x = h\), where \(h\) is the x-coordinate of the vertex which we found to be -1/2. So the axis of symmetry is \(x = -1/2\).

Plot the Graph

Plot the vertex at point \((-1/2, 7/4)\). Draw the axis of symmetry at \(x = -1/2\), which is a vertical line that passes through the vertex. Sketch the graph so it opens upward from the vertex following the \(U\) shape of all parabolas and symmetric with respect to the axis of symmetry.

Key concepts

Quadratic Function

A quadratic function is characterized by an equation in the form of \(y = ax^{2} + bx + c\), where \(a\), \(b\), and \(c\) are constants, and importantly, \(a \eq 0\). This type of function creates a graph known as a parabola, a symmetric curved shape that can either open upwards or downwards depending on the value of \(a\). When graphing a quadratic function, the squared term \(ax^{2}\) is the predominant part, defining the curvature and width of the parabola. If \(a > 0\), the parabola opens upward, like a bowl that holds water. Conversely, if \(a < 0\), it opens downward, resembling an upside-down bowl.

When dealing with exercises involving quadratic functions, always start by identifying the squared term to determine the direction of the opening. In the example \(y = x^{2} + x + 2\), it's evident that \(a = 1\), ensuring the parabola will open upward. This initial step sets the stage for further exploration of the graph's properties, such as the vertex, axis of symmetry, and the exact shape of the curve.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point, depending on whether it opens downward or upward, respectively. For the quadratic function \(y = ax^{2} + bx + c\), the vertex can be calculated using the formula \(h = -b/(2a)\) for the x-coordinate, and then substituting \(h\) back into the equation to find the y-coordinate, \(k\).

In our example \(y = x^{2} + x + 2\), with \(a = 1\) and \(b = 1\), applying the formula yields \(h = -1/(2*1) = -1/2\). Substituting \(h\) into the original equation to find \(k\) gives us \(k = (-1/2)^{2} + (-1/2) + 2 = 7/4\). Therefore, the vertex coordinates are \( (-1/2, 7/4)\). The location of the vertex is crucial since it gives us a reference point around which the entire parabola is shaped.

Axis of Symmetry

The axis of symmetry in a parabolic graph is an imaginary vertical line that neatly divides the parabola into mirror images on either side. It passes through the vertex and each point on the parabola has a corresponding point equidistant on the opposite side of the axis. For the formula \(y = ax^{2} + bx + c\), the equation of the axis of symmetry is always \(x = h\), where \(h\) is the x-coordinate of the vertex.

In the given function \(y = x^{2} + x + 2\), we previously found the vertex to be at \(h = -1/2\). Hence, the equation of the axis of symmetry is \(x = -1/2\). Identifying the axis of symmetry is a key step in graphing any quadratic function as it allows us to ensure that the resulting parabola is graphed with the correct symmetry.

Parabola Opening Direction

The parabola opening direction is determined by the coefficient of the \(x^{2}\) term in a quadratic function, denoted by \(a\) in the general form \(y = ax^{2} + bx + c\). If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards. This is critical to understanding the behavior of the graph, as the opening direction affects the location of the vertex on the graph and the function's minimum or maximum value.

For the function \(y = x^{2} + x + 2\), the value of \(a\) is positive 1, making the parabola open upward. This tells us that the vertex is the lowest point on the graph, known as the minimum point. When graphing, start by plotting the vertex and drawing the parabola with the appropriate opening direction from that point. In this case, sketch the graph with a U-shape, ensuring it's symmetrical about the axis of symmetry and that it rises indefinitely on both sides as it moves away from the vertex.