# Graphing Quadratic Equations in Two Variables

## Graphing Quadratic Equations in Two Variables

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 $$y={x}^{2}-6x+8$$.

### Graph a quadratic equation in two variables.

1. Write the quadratic equation with $$y$$ on one side.
2. Determine whether the parabola opens upward or downward.
3. Find the axis of symmetry.
4. Find the vertex.
5. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
6. Find the x-intercepts.
7. 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.

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