This parabola was created from an equation. What was the equation?

In this concept, you will learn to understand the equation of a parabola.

### Parabolas

A **parabola** is a U-shaped graph.Equations with the ‘\begin{align*}x\end{align*}’ variable raised to the 2^{nd} power are called **quadratic equations **and their graphs are always parabolas.

Here is a quadratic equation.

\begin{align*}y=x^2 -2\end{align*}

The graph of a parabola can change position, direction, and width based on the coefficients of \begin{align*}x^2\end{align*} and \begin{align*}x\end{align*} as well as the constant. Because those pieces of the equation are so important, you name them in what is called the **standard form**.

Standard form of a quadratic equation: \begin{align*}y=ax^2 +bx+c\end{align*} (where ‘\begin{align*}a\end{align*}’ cannot be zero). Notice that ‘\begin{align*}a\end{align*}’ and ‘\begin{align*}b\end{align*}’ are coefficients and can be either positive or negative. The value of ‘\begin{align*}c\end{align*}’ is a constant. All of these values affect the parabola that is graphed.

Once again, the ‘\begin{align*}a\end{align*}’ value can predict two things:

1. how wide the graph will be

Generally speaking, the further the ‘\begin{align*}a\end{align*}’ value is from zero, the narrower the graph; the closer the ‘\begin{align*}a\end{align*}’ value is to zero, the wider the graph.

2. if the graph opens upward or downward.

A positive value of ‘\begin{align*}a\end{align*}’ will give a graph that opens upwards while a negative value of ‘\begin{align*}a\end{align*}’ will give a graph that opens downwards.

What about the ‘\begin{align*}b\end{align*}’ value?

All of the parabolas are symmetrical—they are the same on both sides, as if they were reflected on a mirror that were right down the middle of the graph. This reflection line is called the **axis of symmetry**. The ‘\begin{align*}b\end{align*}’ value helps us to predict the axis of symmetry.

Finally, the \begin{align*}c\end{align*} value, determines the \begin{align*}y\end{align*}-intercept of the graph—it tells where the graph will cross the \begin{align*}y\end{align*}-axis. When the \begin{align*}c\end{align*} value was 3, the graph crossed the \begin{align*}y\end{align*}-axis at 3.

Let’s look at some graphs.

Looking at these graphs, and knowing what the \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*} values of the quadratic equation represent, will help to determine the equation of the graph. Here is a chart to help you understand what you can determine by these graphs.

Now you can see how the graphs of each equation provide you with information.

You have learned to write linear equations based on linear graphs, you can also find a quadratic equation by using the parabola.

You know that the ‘\begin{align*}a\end{align*}’ value tells if the graph goes upward or downward. So, if the graph goes downward, the ‘\begin{align*}a\end{align*}’ value must be negative. If the graph opens upward, the ‘\begin{align*}a\end{align*}’ value must be positive.

You also know that the ‘\begin{align*}c\end{align*}’ value tells you the \begin{align*}y\end{align*}-intercept on the graph. So, if you know the \begin{align*}y\end{align*}-intercept, then you know the ‘\begin{align*}c\end{align*}’ value.

If you have a graph, then you can also work backwards. In other words, you can fill in a t-table using the points you see on the graph. Then, by looking for a pattern in the t-table, you can derive the equation.

Let’s look at an example.

Write the equation for the given graph.

First, start with what you know about the values of ‘\begin{align*}a\end{align*}’ and ‘\begin{align*}c\end{align*}’.

\begin{align*}a\end{align*}: Graph opens downward so \begin{align*}a < 0\end{align*}.

\begin{align*}c\end{align*}: The \begin{align*}y\end{align*}-intercept is (0, 3) so \begin{align*}c = 3\end{align*}.

Next, construct a table of values from the graph.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |

-2 | -1 |

-1 | 2 |

0 | 3 |

1 | 2 |

2 | -1 |

Then, put what you know into the standard form of the quadratic equation. Since the graph goes over one and down 1, you know that \begin{align*}a = -1\end{align*}.

\begin{align*}\begin{array}{rcl} y &=& ax^2 + bx + c \\ y &=& -x^2 + bx + 3 \end{array}\end{align*}

Then, test a point on the graph to find the value of \begin{align*}b\end{align*}.

\begin{align*}y = x^2 + bx + 3\end{align*}

Point: (-1, 2)

\begin{align*}\begin{array}{rcl} 2 &=& - (-1)^2 + b(-1) + 3 \\ 2 &=& -1 +-b + 3 \\ 2 &=& -b +2 \\ b &=& 0 \end{array}\end{align*}

The answer is \begin{align*}y=-x^2+3\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about the parabola.

First, start with what you know about the values of ‘\begin{align*}a\end{align*}’ and ‘\begin{align*}c\end{align*}’.

\begin{align*}a\end{align*}: Graph opens upward so \begin{align*}a > 0\end{align*}.

\begin{align*}c\end{align*}: The \begin{align*}y\end{align*}-intercept is (0, 0) so \begin{align*}c = 0\end{align*}.

Next, construct a table of values from the graph.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |

-2 | 4 |

-1 | 1 |

0 | 0 |

1 | 1 |

2 | 4 |

Then, put what you know into the standard form of the quadratic equation. Since the graph goes over one and up 1, you know that \begin{align*}a = 1\end{align*}.

\begin{align*}\begin{array}{rcl} y &=& ax^2 + bx + c \\ y &=& x^2 + bx \end{array}\end{align*}

Then, test a point on the graph to find the value of \begin{align*}b\end{align*}.

\begin{align*}y = -x^2 + bx\end{align*}

Point: (-1, 1)

\begin{align*}\begin{array}{rcl} 1 &=& (-1)^2 + b(-1) \\ 1 &=& 1+ -b \\ b &=& 0 \end{array} \end{align*}

The answer is \begin{align*}y=x^2\end{align*}.

#### Example 2

Figure out the equation for the following parabola.

First, start with what you know about the values of ‘\begin{align*}a\end{align*}’ and ‘\begin{align*}c\end{align*}’.

\begin{align*}a\end{align*}: Graph opens upward so \begin{align*}a > 0\end{align*}.

\begin{align*}c\end{align*}: The \begin{align*}y\end{align*}-intercept is (0, -4) so \begin{align*}c = -4\end{align*}.

Next, construct a table of values from the graph.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |

-2 | 0 |

-1 | -3 |

0 | -4 |

1 | -3 |

2 | 0 |

Then, put what you know into the standard form of the quadratic equation. Since the graph goes over one and up 1, you know that \begin{align*}a = 1\end{align*}.

\begin{align*}\begin{array}{rcl} y &=& ax^2 + bx + c \\ y &=& x^2 + bx - 4 \end{array}\end{align*}

Then, test a point on the graph to find the value of \begin{align*}b\end{align*}.

\begin{align*}y= -x^2 + bx - 4\end{align*}

Point: (-1, -3)

\begin{align*}\begin{array}{rcl} -3 &=& (-1)^2 + b (-1) -4 \\ -3 &=& 1 + -b -4 \\ -3 &=& -b -3 \\ b &=& 0 \end{array}\end{align*}

The answer is \begin{align*}y=x^2-4\end{align*}.

#### Example 3

If the \begin{align*}c\end{align*} value is 4, where is the \begin{align*}y\end{align*}-intercept of the graph?

If \begin{align*}c = 4\end{align*}, the \begin{align*}y\end{align*}–intercept or the point where the curve crosses the \begin{align*}y\end{align*}-axis is (0, 4).

#### Example 4

If the ‘\begin{align*}a\end{align*}’ value is −3, will the parabola open upward or downward?

If \begin{align*}a < 0\end{align*}, then the graph opens downward so when \begin{align*}a = -3\end{align*}, the graph will open downward.

#### Example 5

If the parabola opens upward, which value is positive \begin{align*}a, b\end{align*} or \begin{align*}c\end{align*}.

When the graph opens upward or downward, the value of ‘\begin{align*}a\end{align*}’ is affected. Therefore if the graph opens upward you know that ‘\begin{align*}a\end{align*}’ is positive.

### Review

Answer the following questions about parabolas.

1. True or false. All parabolas are symmetrical.

2. True or false. The \begin{align*}y\end{align*} intercept is the same as the \begin{align*}c\end{align*} value.

3. A parabola with a positive squared value opens __________.

4. A parabola with a negative squared value opens __________.

5. What is the vertex of the parabola?

6. True or false. A parabola always forms a U shape.

7. True or false. The closer the \begin{align*}a\end{align*} value is to zero the wider the parabola.

8. True or false. The closer the \begin{align*}a\end{align*} value is to zero the narrower the parabola.

9. True or false. The \begin{align*}b\end{align*} value determines the axis of symmetry.

10. What does the \begin{align*}c\end{align*} value indicate?

11. True or false. A linear equation will have a graph that is a parabola.

12. True or false. A quadratic equation and a linear equation will have a similar graph.

Write the equations of the following graphs. Use the \begin{align*}c\end{align*} values and a t-table to help you.

and13.

14.

15.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.13.