Problem 2 - Algebra 2

Question

Graph each function. $$ y=-x^{2}-1 $$

Step-by-Step Solution

Answer

The graph of the quadratic function \(y = -x^{2} - 1\) is a downward-opening parabola with the vertex at the point \((0, -1)\).

Step 1: Identify the type of function

Recognize that the function given is a quadratic function in the standard form of \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients. In this equation, \(a = -1\), \(b = 0\), and \(c = -1\). The negative value of \(a\) indicates that the parabola opens downward.

Step 2: Determine the vertex

The vertex of a parabola in standard form \(y = ax^2 + bx + c\) can be found using the formula \(h = -\frac{b}{2a}\) for the x-coordinate. For the function \(y = -x^2 - 1\), since \(b = 0\), the x-coordinate of the vertex, \(h\), is \(0\). The y-coordinate is found by plugging \(h\) into the function: \(k = -h^2 - 1 = -1\). Thus, the vertex of the parabola is \((0, -1)\).

Step 3: Plot the vertex

Begin the graph by plotting the vertex \((0, -1)\) on the coordinate plane. This point is the highest point on the graph because the parabola opens downward.

Step 4: Find additional points

Choose values for \(x\) to the left and right of the vertex to find corresponding \(y\) values and plot these points. For example, if \(x = 1\), then \(y = -1^2 - 1 = -2\), and if \(x = -1\), then \(y = -(-1)^2 - 1 = -2\). Plot the points \((1, -2)\) and \((-1, -2)\). You can find more points in a similar manner to get a more accurate graph.

Step 5: Draw the parabola

Draw a smooth curve through the plotted points, ensuring that the graph is a mirror image on either side of the vertex since parabolas are symmetrical. Extend the curve downwards in both directions, maintaining its shape as it moves further away from the vertex.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that is characterized by having the highest degree of 2. It can be expressed in the general form of \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants with \(a\) not equal to zero.

When graphing a quadratic function, you'll notice it creates a U-shaped curve called a parabola. Depending on the sign of the coefficient \(a\), the parabola opens upwards if \(a\) is positive or downwards if \(a\) is negative, as seen in our exercise with \(y = -x^2 - 1\), which opens downward. The graph's concavity is determined by the sign of \(a\), so in our case, the parabola will be concave down because \(a = -1\).

The beauty of quadratic functions lies in their symmetry. This property of symmetry can be used to plot points on one side of the parabola and mirror them on the other side to get an accurate representation of the function.

Vertex of a Parabola

The vertex of a parabola is the point where it turns; thus, it is either the lowest or highest point on the graph, depending on whether the parabola opens up or down. To find the vertex of the parabola represented by the standard form equation \(y = ax^2 + bx + c\), one can use the formula \(h = -\frac{b}{2a}\) to find the x-coordinate.

For our exercise function \(y = -x^2 - 1\), with \(a = -1\) and \(b = 0\), we calculated the vertex to be at \(h = 0\), which makes the y-coordinate \(k = -1\), giving us a vertex at \((0, -1)\). This is a notable feature because the vertex is a crucial point in sketching the graph of the function. It serves as a 'guide post' around which you draw the rest of the parabola. Knowing the vertex allows us to plot the axis of symmetry of the parabola, which is the vertical line that passes through the vertex, and is essential for understanding the overall shape and direction of the graph.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is the most commonly used form to represent quadratic functions. It is written as \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients and \(a\) cannot be zero. This form is particularly useful because it clearly shows whether the parabola opens upwards or downwards through the coefficient \(a\).

In addition, the standard form makes it relatively simple to calculate the vertex, axis of symmetry, and the y-intercept of the parabola. The y-intercept is simply the value of \(c\), the constant term, because it's the point where the parabola crosses the y-axis (when \(x = 0\)). This form is also conducive to applying the quadratic formula for finding the roots of a quadratic equation when setting \(y = 0\). By analyzing the coefficients in standard form, one can also determine the parabola's width and orientation and predict the general shape of the graph without plotting any points.