# What are the important points needed to graph y= 2(x-3)^2+4?

Jul 6, 2018

Refer to the explanation.

#### Explanation:

Graph:

$y = 2 {\left(x - 3\right)}^{2} + 4$ is a quadratic equation in vertex form:

$y = a {\left(x - h\right)}^{2} + k$,

where:

$a = 2$, $h = 3$, $k = 4$

The important points to graph a parabola are the axis of symmetry, vertex, the y-intercept, x-intercepts (if there are real solutions), and additional points (especially if there are no x-intercepts).

Axis of symmetry: vertical line that divides the parabola into two equal halves. In vertex form, the axis of symmetry is:

$x = h$

$x = 3$

Vertex: maximum or minimum point of the parabola

Since $a > 0$, the vertex is the minimum point and the parabola opens upward.

The vertex is $\left(h , k\right)$, which in this equation is $\left(3 , 4\right)$. Plot this point.

Because the vertex $\left(3 , 4\right)$ is above the x-axis, and it is the minimum point, there are no x-intercepts for this equation, so you will need additional points to graph the parabola.

Y-intercept: value of $y$ when $x = 0$

Substitute $0$ for $x$ and solve for $y$.

$y = 2 {\left(0 - 3\right)}^{2} + 4$

$y = 2 {\left(- 3\right)}^{2} + 4$

$y = 2 \left(9\right) + 4$

$y = 18 + 4$

$y = 22$

The y-intercept is $\left(0 , 22\right)$. Plot this point.

To determine additional points, choose values for $x$ and solve for $y$.

Since the axis of symmetry is $x = 3$, you can use this to determine symmetrical points. For example, the y-intercept is $\left(0 , 22\right)$ which is three spaces to the left of $3$, so the symmetrical point is three spaces farther than $3$, or $x = 6$.

Additional point 1: $x = 6$

$y = 2 {\left(6 - 3\right)}^{2} + 4$

$y = 2 {\left(3\right)}^{2} + 4$

$y = 2 \left(9\right) + 4$

$y = 18 + 4$

$y = 22$

Additional point 1: $\left(6 , 22\right)$ Plot this point.

Additional point 2: $x = 1$

$y = 2 {\left(1 - 3\right)}^{2} + 4$

$y = 2 {\left(- 2\right)}^{2} + 4$

$y = 2 \left(4\right) + 4$

$y = 12$

Additional point 2: $\left(1 , 12\right)$ Plot this point.

Additional point 3: $x = 5$

$y = 2 {\left(5 - 3\right)}^{2} + 4$

$y = 2 {\left(2\right)}^{2} + 4$

$y = 2 \left(4\right) + 4$

$y = 12$

Additional point 3: $\left(5 , 12\right)$ Plot this point.

Sketch a parabola through the points. Do not connect the dots.

graph{y=2(x-3)^2+4 [-9.33, 10.67, 22.08, 32.08]}