Question

Graph each function. Set the viewing window for \(x\) and \(y\) initially from -5 to 5 then resize if needed. $$y=x^{2}$$

Short answer

The graph of \(y = x^2\) is a parabola opening upwards with the vertex at the origin. Start with a window of -5 to 5 for both axes and resize if necessary.

Step-by-step solution

Understand the Graph of a Quadratic Function

Recognize that the function given, \(y = x^2\), represents a quadratic function and its graph is a parabola. The most important features of this graph are its vertex, axis of symmetry, and the direction in which it opens. In this case, the vertex is at the origin \((0, 0)\), the axis of symmetry is the y-axis \(x = 0\), and the parabola opens upwards.

Choose Initial Viewing Window

Setup the graph with the initial viewing window for both \(x\) and \(y\) values ranging from -5 to 5. This window is a standard starting point to analyze the basic shape and features of the given quadratic function.

Plot Points on the Graph

Calculate and plot points for the function within the viewing window. Since the graph is symmetrical around the y-axis, you only need to calculate the function values for non-negative or non-positive values of \(x\) and then mirror them across the y-axis. For example, when \(x = -2, 0, 2\), the corresponding \(y\) values from the function \(y = x^2\) would be \(4, 0, 4\) respectively.

Draw the Parabola

Using the points plotted, draw a smooth curve to form the parabola. Make sure that the parabola is symmetrical around the y-axis and its vertex is at the origin as determined in Step 1.

Resize Viewing Window if Needed

Examine the graph to see if the entire parabola or its relevant portions are visible within the initial viewing window. Resize the window if portions of the graph are not visible or if you need to show more details.

Key concepts

Parabola Graph

When graphing a quadratic function, which is any function that can be expressed in the form \(y = ax^2 + bx + c\), the shape of the graph will always be a parabola. A parabola graph is a mirror-symmetrical, U-shaped curve that may open upwards or downwards

For the quadratic equation \(y = x^2\), as seen in the exercise, the parabola opens upwards because the coefficient of \(x^2\) is positive. We begin our graphing by choosing an appropriate scale for the x and y axes, typically starting with -5 to 5 for both. By plotting calculated points, such as \((0, 0)\), \((1, 1)\), \((-1, 1)\), and so forth, we get a visual representation of the function's behavior. The points should fall into a symmetrical 'U' shape indicating we have correctly graphed our parabola.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. In our example function \(y = x^2\), the vertex is at the origin \((0, 0)\).

This is because the vertex form of a quadratic function is \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex. With no \(h\) or \(k\) values present in \(y = x^2\), the vertex defaults to \(0, 0\) which is the starting point for our parabola. The vertex is crucial for graphing because it provides a clear starting point from which the parabola shape is defined.

Axis of Symmetry

The axis of symmetry in a quadratic function is a vertical line that splits the parabola into two mirror images, giving it its symmetrical nature. For the given function \(y = x^2\), the axis of symmetry is the y-axis, represented by the equation \(x = 0\).

This line passes directly through the vertex and is a useful tool when graphing because we can reflect points from one side of the parabola to the other across the axis of symmetry. To guarantee accuracy when plotting additional points, one can always calculate points on one side of the parabola and use the axis of symmetry to find their counterparts on the opposite side.

Quadratic Function Properties

Quadratic functions exhibit several distinct properties that help us understand and graph them accurately. The standard form of a quadratic function is \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

Some crucial properties of quadratic functions include:

• The direction in which a parabola opens is determined by the coefficient \(a\): if \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards.
• The vertex represents the function's maximum or minimum value, depending on the direction it opens.
• The y-intercept is the point where the graph crosses the y-axis, and in the form \(y = ax^2 + bx + c\), it is represented by \(c\).
• The x-intercepts, or zeros of the function, are points where the graph crosses the x-axis. These can be found by solving \(ax^2 + bx + c = 0\).

Understanding these properties not only makes graphing easier but also aids in comprehending the overall behavior of quadratic functions.