Question

graph the parabola.

$y = 2(x - 5)^2-3$

plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex. then click on the graph - a - function button.

Explanation:

Step1: Identify the vertex form of parabola

The vertex - form of a parabola is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex. For the equation $y = 2(x - 5)^2-3$, $a = 2$, $h = 5$, $k=-3$. So the vertex is $(5,-3)$.

Step2: Find points to the left of the vertex

Let's choose $x = 3$ and $x = 4$.
When $x = 3$:
$y=2(3 - 5)^2-3=2\times(-2)^2-3=2\times4-3=8 - 3=5$. So the point is $(3,5)$.
When $x = 4$:
$y=2(4 - 5)^2-3=2\times(-1)^2-3=2\times1-3=-1$. So the point is $(4,-1)$.

Step3: Find points to the right of the vertex (by symmetry)

Since the parabola $y=a(x - h)^2+k$ is symmetric about the line $x = h$.
When $x = 6$ (1 unit to the right of the vertex, as $x = 4$ is 1 unit to the left), $y=2(6 - 5)^2-3=2\times1^2-3=2 - 3=-1$. So the point is $(6,-1)$.
When $x = 7$ (2 units to the right of the vertex, as $x = 3$ is 2 units to the left), $y=2(7 - 5)^2-3=2\times2^2-3=8 - 3=5$. So the point is $(7,5)$.

Answer:

The five points are $(3,5),(4,-1),(5,-3),(6,-1),(7,5)$