Graphing Quadratic Equations: example 10.51

Graph $y = x^{2} + 4 x + 5$ .

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Graph the parabola $y = 2 x^{2} - 6 x + 5 .$

How did it go?

Graph $y = x^{2} + 4 x + 5$ .

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Graph the parabola $y = 2 x^{2} - 6 x + 5 .$

How did it go?

The equation has y on one side.
Since a is 1, the parabola opens upward.
To find the axis of symmetry, find $x = - \frac{b}{2 a} .$		The axis of symmetry is $x = −2 .$
The vertex is on the line $x = −2 .$
Find y when $x = −2 .$		The vertex is $(−2, 1) .$
The y-intercept occurs when $x = 0 .$ Substitute $x = 0 .$ Simplify. The point $(0, 5)$ is two units to the right of the line of symmetry. The point two units to the left of the line of symmetry is $(−4, 5) .$		The y-intercept is $(0, 5) .$ Point symmetric to the y- intercept is $(−4, 5)$ .
The x- intercept occurs when $y = 0 .$
Substitute $y = 0 .$ Test the discriminant.
		$b^{2} - 4 a c$ $4^{2} - 4 \cdot 15$ $16 - 20$ $−4$
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.

The equation has y on one side.
Since a is 1, the parabola opens upward.
To find the axis of symmetry, find $x = - \frac{b}{2 a} .$		The axis of symmetry is $x = −2 .$
The vertex is on the line $x = −2 .$
Find y when $x = −2 .$		The vertex is $(−2, 1) .$
The y-intercept occurs when $x = 0 .$ Substitute $x = 0 .$ Simplify. The point $(0, 5)$ is two units to the right of the line of symmetry. The point two units to the left of the line of symmetry is $(−4, 5) .$		The y-intercept is $(0, 5) .$ Point symmetric to the y- intercept is $(−4, 5)$ .
The x- intercept occurs when $y = 0 .$
Substitute $y = 0 .$ Test the discriminant.
		$b^{2} - 4 a c$ $4^{2} - 4 \cdot 15$ $16 - 20$ $−4$
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.