Questions: Graph the parabola. y=3x^2-30x+70 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.

Graph the parabola.
y=3x^2-30x+70

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.

Transcript text: Graph the parabola. \[ y=3 x^{2}-30 x+70 \] 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.

Solution

Solution Steps

Step 1: Find the vertex.

The x-coordinate of the vertex is given by $x = -\frac{b}{2a}$ . In the given equation, $y = 3x^2 - 30x + 70$ , $a = 3$ and $b = -30$ . So, $x = -\frac{-30}{2(3)} = \frac{30}{6} = 5$ .

To find the y-coordinate of the vertex, substitute $x = 5$ into the equation: $y = 3(5)^2 - 30(5) + 70 = 3(25) - 150 + 70 = 75 - 150 + 70 = -5$ . So, the vertex is $(5, -5)$ .

Step 2: Find two points to the left of the vertex.

Let's choose $x = 3$ and $x = 4$ . When $x = 3$ , $y = 3(3)^2 - 30(3) + 70 = 27 - 90 + 70 = 7$ . So, the point is $(3, 7)$ . When $x = 4$ , $y = 3(4)^2 - 30(4) + 70 = 48 - 120 + 70 = -2$ . So, the point is $(4, -2)$ .

Step 3: Find two points to the right of the vertex.

Let's choose $x = 6$ and $x = 7$ . When $x = 6$ , $y = 3(6)^2 - 30(6) + 70 = 108 - 180 + 70 = -2$ . So, the point is $(6, -2)$ . When $x = 7$ , $y = 3(7)^2 - 30(7) + 70 = 147 - 210 + 70 = 7$ . So, the point is $(7, 7)$ .

Final Answer

The vertex is $(5, -5)$ . Two points to the left of the vertex are $(3, 7)$ and $(4, -2)$ . Two points to the right of the vertex are $(6, -2)$ and $(7, 7)$ .