Question

Graph the parabola. y = - 3x ^ 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.

Answer 1

Step-by-step explanation:

The equation we have is:

`y=-3x^2+3`

And we are asked to plot the parabola by finding 5 points on the graph.

Step 1. Finding the vertex of the parabola.

To find the vertex, we rewrite the equation in the vertex form:

`y=a\left(x-h\right)^2+k`

where (h, k) is the vertex.

Rewriting our equation:

`\begin{gathered} y=-3x^(2)+3 \\ \downarrow\downarrow \\ y=-3\left(x-0\right)^2+3 \end{gathered}`

By comparing this with the vertex form, we find the values of h and k for the vertex:

`\begin{gathered} h=0 \\ k=3 \\ vertex\text{ }\left(h,k\right):\text{ }\left(0,3\right) \end{gathered}`

The first point and the vertex of the parabola is at (0,3).

Step 2. We also need to find two points to the left of the vertex and two points to the right. In this step, we will find the two points to the left.

x=-1 is at the left if x=0 (where the vertex is) so we substitute x=-1 into the equation to find the corresponding y-value and form an (x,y) point:

`\begin{gathered} Using\text{ x=-1} \\ y=-3x^(2)+3 \\ Substituting\text{ x=-1 as the x-value} \\ y=-3\left(-1\right)^2+3 \\ Solving\text{ the operations:} \\ y=-3\left(1\right)+3 \\ y=-3+3 \\ y=0 \\ The\text{ point is \lparen x,y\rparen:} \\ \lparen-1,0) \end{gathered}`

We do the same now but for another x value to the left of x=0, the value we will take is x=-2:

`\begin{gathered} Using\text{ x=}-2 \\ y=-3\left(-2\right)^2+3 \\ y=-3\left(4\right)+3 \\ y=-12+3 \\ y=-9 \\ Point\text{ \lparen x,y\rparen is at \lparen-2,-9\rparen} \end{gathered}`

Step 3. To find two points to the right, we use x=1 and x=2.

For x=1:

`\begin{gathered} using\text{ x=1} \\ y=-3\left(1\right)^2+3 \\ y=-3+3 \\ y=0 \\ point\text{ \lparen x,y\rparen is at \lparen1,0\rparen} \\ using\text{ x=2} \\ y=-3(2)^2+3 \\ y=-3(4)^+3 \\ y=-12+3 \\ y=-9 \\ Pont\text{ \lparen x,y\rparen is at \lparen2,-9\rparen} \end{gathered}`

Step 4. Plotting all of the points:

The graph is:

Answer:

(0,3)

(-1,0)

(1,0)

(-2,-9)

(2,-9)