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 \((7, 7)\).

The five points are \((3,7)\), \((4,-2)\), \((5,-5)\), \((6,-2)\), and \((7,7)\). \( \boxed{ (5, -5), (4, -2), (3, 7), (6, -2), (7, 7)} \)

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