Graphing Equations in Standard Form
The equation that is provided to you will determine what information you need to analyze in order to create a graph of a parabola. Quadratic equations in both standard form and vertex form provide necessary information to plot the vertex of the parabola, although in different ways.
The standard form of a quadratic equation is:
The coefficient in front of the x-squared term, a, will tell you if the parabola opens upwards of downward. If a is a positive value, then the parabola opens upwards, and will have a U-shape to it. This means that the vertex is a minimum point. If a is negative, then the parabola will open downward, with an upside down U-shape to it. This means that the vertex is a maximum point to the parabola.
The vertex lies on the parabola's axis of symmetry, which is a line of reflection that cuts the parabola in half. This helps us graph parabolas, because if we know this axis of symmetry, we can plot a few points on one side of the parabola, and then just reflect them over to find points on the other side. With only a few points plotted on each side of the axis of symmetry, we can draw the general shape of the parabola.
To find the axis of symmetry with equations in standard form, we use this formula:
where a and b come from the coefficients of the x-squared and x-terms in standard form.
Let's find the equation to the axis of symmetry for the following equation:
So far, we know that the axis of symmetry is the line x = –1. This means that the vertex of the parabola is somewhere on the line. In order to find its y–coordinate, we can plug –1 into the original equation, and solve for y. Then we will be able to plot the vertex on the graph:
The vertex to the parabola is at the point (–1, 4). Let's sketch the graph so far:
Now, we just need to figure out a few points to be plotted on one side of the parabola. Then we'll be able to reflect about the line of symmetry to complete our graph. Let's choose some x–values on the left side of the axis of symmetry.
Here is the sketch of the graph with these points plotted, as well as reflected about the axis of symmetry:
Graphing Equations in Vertex Form
When equations are provided in vertex form, we still want to locate the parabola's vertex, plot a few points on one side of the axis of symmetry, and reflect them to plot a more complete graph. However, finding the coordinates of the vertex using this equation is much easier. The only tricky thing is to keep the sign in mind.
Vertex form of a quadratic equation is:
The variables h and k describe the x– and y–coordinates of the vertex: (h, k)
Notice that generally, there is a minus sign between x and h. So if the equation has a plus sign between x and h, the value of h is negative, rather than positive. This can be counter intuitive, so keep this in mind.
Let's graph the equation:
We can identify the vertex right away from equations in this form. The vertex of this parabola is located at the point (–2, 1). Remember, h has a value of –2, since the equation contains (x + 2). We take the sign of k with us, so we know that the y–coordinate remains positive.
We can now graph the parabola's vertex, along with its axis of symmetry:
Similarly, we will choose a few x-values on one side of the axis of symmetry, and find their associated y–values. We can then plot points and reflect them about the axis of symmetry to create a nice sketch of the parabola: