## Graphing Quadratic Equations in Two Variables

Contents

Now, we have all the pieces we need in order to graph a quadratic equation in two variables. We just need to put them together. In the next example, we will see how to do this.

## Example: How To Graph a Quadratic Equation in Two Variables

Graph \(y={x}^{2}-6x+8\).

### Solution

### Graph a quadratic equation in two variables.

- Write the quadratic equation with \(y\) on one side.
- Determine whether the parabola opens upward or downward.
- Find the axis of symmetry.
- Find the vertex.
- Find the
*y*-intercept. Find the point symmetric to the*y*-intercept across the axis of symmetry. - Find the
*x*-intercepts. - Graph the parabola.

We were able to find the *x*-intercepts in the last example by factoring. We find the *x*-intercepts in the next example by factoring, too.

## Example

Graph \(y=\text{−}{x}^{2}+6x-9\).

### Solution

The equation y has on one side. | ||

Since a is \(-1\), the parabola opens downward.To find the axis of symmetry, find \(x=-\frac{b}{2a}\). | The axis of symmetry is \(x=3.\) The vertex is on the line \(x=3.\) | |

Find y when \(x=3.\) | The vertex is \(\left(3,0\right).\) | |

The y-intercept occurs when \(x=0.\)Substitute \(x=0.\) Simplify. The point \(\left(0,-9\right)\) is three units to the left of the line of symmetry. The point three units to the right of the line of symmetry is \(\left(6,-9\right).\) Point symmetric to the y-intercept is \(\left(6,-9\right)\) | The y-intercept is \(\left(0,-9\right).\) | |

The x-intercept occurs when \(y=0.\) | ||

Substitute \(y=0.\) | ||

Factor the GCF. | ||

Factor the trinomial. | ||

Solve for x. | ||

Connect the points to graph the parabola. |

For the graph of \(y=-{x}^{2}+6x-9\), the vertex and the *x*-intercept were the same point. Remember how the discriminant determines the number of solutions of a quadratic equation? The discriminant of the equation \(0=\text{−}{x}^{2}+6x-9\) is 0, so there is only one solution. That means there is only one *x*-intercept, and it is the vertex of the parabola.

How many *x*-intercepts would you expect to see on the graph of \(y={x}^{2}+4x+5\)?

## Example

Graph \(y={x}^{2}+4x+5\).

### Solution

The equation has y on one side. | ||

Since a is 1, the parabola opens upward. | ||

To find the axis of symmetry, find \(x=-\frac{b}{2a}.\) | The axis of symmetry is \(x=-2.\) | |

The vertex is on the line \(x=-2.\) | ||

Find y when \(x=-2.\) | The vertex is \(\left(-2,1\right).\) | |

The y-intercept occurs when \(x=0.\)Substitute \(x=0.\) Simplify. The point \(\left(0,5\right)\) is two units to the right of the line of symmetry. The point two units to the left of the line of symmetry is \(\left(-4,5\right).\) | The y-intercept is \(\left(0,5\right).\)Point symmetric to the y- intercept is \(\left(-4,5\right)\). | |

The x– intercept occurs when \(y=0.\) | ||

Substitute \(y=0.\) Test the discriminant. | ||

\({b}^{2}-4ac\) \({4}^{2}-4\cdot 15\) \(16-20\) \(\phantom{\rule{1em}{0ex}}-4\) | ||

Since the value of the discriminant is negative, there is no solution and so no x- intercept.Connect the points to graph the parabola. You may want to choose two more points for greater accuracy. |

Finding the *y*-intercept by substituting \(x=0\) into the equation is easy, isn’t it? But we needed to use the Quadratic Formula to find the *x*-intercepts in the example above. We will use the Quadratic Formula again in the next example.

## Example

Graph \(y=2{x}^{2}-4x-3\).

### Solution

The equation y has one side.Since a is 2, the parabola opens upward. | ||

To find the axis of symmetry, find \(x=-\frac{b}{2a}\). | The axis of symmetry is \(x=1\). | |

The vertex on the line \(x=1.\) | ||

Find y when \(x=1\). | The vertex is \(\left(1,\text{−}5\right)\). | |

The y-intercept occurs when \(x=0.\) | ||

Substitute \(x=0.\) | ||

Simplify. | The y-intercept is \(\left(0,-3\right)\). | |

The point \(\left(0,-3\right)\) is one unit to the left of the line of symmetry. The point one unit to the right of the line of symmetry is \(\left(2,-3\right)\) | Point symmetric to the y-intercept is \(\left(2,-3\right).\) | |

The x-intercept occurs when \(y=0\). | ||

Substitute \(y=0\). | ||

Use the Quadratic Formula. | ||

Substitute in the values of a, b, c. | ||

Simplify. | ||

Simplify inside the radical. | ||

Simplify the radical. | ||

Factor the GCF. | ||

Remove common factors. | ||

Write as two equations. | ||

Approximate the values. | ||

The approximate values of the x-intercepts are \(\left(2.5,0\right)\) and \(\left(-0.6,0\right)\). | ||

Graph the parabola using the points found. |