Question

Sketch the graph of the function. $$y=x^{2}+x+2$$

Short answer

The graph of the function \(y = x^2 + x + 2\) is a parabola opening upwards with its vertex at \((-0.5, 1.75)\) and axis of symmetry at \(x=-0.5\).

Step-by-step solution

Calculate the Axis of Symmetry

The formula to find the axis of symmetry of a parabola is \(x = -b/2a\). Plugging in the values from the given function, \(x = -1/(2*1) = -0.5\). The axis of symmetry is at \(x = -0.5\).

Find the Vertex

The vertex form of a parabola is \(y=a(x-h)^2+k\), where \((h,k)\) is the vertex. In this case, \(h\) is the axis of symmetry, so plug \(-0.5\) into the equation \(y = x^{2}+x+2\) to get \(y\). So, \(y = (-0.5)^2 + (-0.5) + 2 = 1.75\). Therefore, the vertex of the parabola is \(-0.5, 1.75\).

Determine the Direction of the Parabola

Because \(a = 1\) is greater than zero in our function \(y = x^2 + x + 2\), the graph will open upwards. Negative \(a\) values would indicate the parabola opens downward.

Sketch the graph

Plot the vertex \((-0.5, 1.75)\), draw an axis of symmetry at \(x=-0.5\). Since the parabola opens upward, draw a U-shaped curve recognizing these points and the axis of symmetry. Pick other points like \(x=-1\) and \(x=0\) to determine the width of the parabola. Calculate \(y\) for these points and plot them.

Key concepts

Axis of Symmetry

The axis of symmetry is a fundamental concept when graphing quadratic functions. It is a vertical line that divides the parabola into two mirror-image halves. This axis is particularly useful because it helps us locate the vertex of the parabola and provides a structured approach to plotting the graph.

For a quadratic function in standard form, which is expressed as \(y = ax^{2} + bx + c\), the axis of symmetry can be calculated using the formula \(x = -\frac{b}{2a}\). In the given exercise, substituting the values from the function \(y = x^{2} + x + 2\), we find the axis of symmetry at \(x = -0.5\). This not only aids in finding the vertex but also ensures that every point on one side of the axis has a corresponding point with the same distance on the other side.

Vertex of a Parabola

The vertex of a parabola represents the highest or lowest point on the graph, depending on its orientation. As the turning point, it determines the parabola's maximum or minimum value.

The vertex can be efficiently found once we have the axis of symmetry. Given the standard form equation of a parabola, we plug the axis value into the function to find the y-coordinate. In our example with the function \(y = x^{2} + x + 2\), substituting \(x = -0.5\) delivers the y-coordinate \(y = 1.75\). As a result, the vertex of the parabola is at \((-0.5, 1.75)\). This crucial point serves as a guide for graphing the entire parabola and indicates the location of the parabola's peak (if it opens downward) or trough (if it opens upward).

Parabola Opening Direction

For a quadratic function, the parabola's opening direction is determined by the coefficient of the \(x^{2}\) term, denoted as 'a'. When the coefficient \(a\) is positive, the parabola opens upward, resembling a 'U' shape. Conversely, if 'a' is negative, the parabola opens downward, similar to an upside-down 'U'.

In the function \(y = x^{2} + x + 2\), since \(a = 1\) is positive, we ascertain that the parabola opens upwards. It means the vertex is the lowest point on the graph, and the parabola will continue to rise infinitely as it moves away from the vertex on either side. This key piece of information helps us sketch the graph accurately and understand the nature of the quadratic function's graph. It is crucial to note this directionality as it affects the shape and the range of the function.