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Draw a graph of the equation for the line $y=3x^2-7$

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Find the vertex of the transformed function

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Q. Draw a graph of the equation for the line

y=3x^2-7

Identify Equation Type: Identify the type of equation we are dealing with. $\newline$ The equation $y=3x^2-7$ is a quadratic equation in the form $y=ax^2+bx+c$ , where $a=3$ , $b=0$ , and $c=-7$ .
Determine Parabola Vertex: Determine the vertex of the parabola. $\newline$ Since the equation is in the form $y=ax^2+bx+c$ and there is no $x$ term, the vertex occurs at $x=0$ . To find the $y$ -coordinate of the vertex, substitute $x=0$ into the equation. $\newline$ $y = 3(0)^2 - 7$ $\newline$ $y = -7$ $\newline$ So, the vertex of the parabola is at $(0, -7)$ .
Plot Vertex on Graph: Plot the vertex on the graph. $\newline$ The vertex $(0, -7)$ is the lowest point of the parabola since $a=3$ is positive, indicating that the parabola opens upwards. Plot the point $(0, -7)$ on the graph.
Find Additional Points: Find additional points to plot on the graph. $\newline$ Choose values for $x$ and calculate the corresponding $y$ values using the equation $y=3x^2-7$ . It's often helpful to choose both positive and negative values for $x$ to get a symmetrical view of the parabola around the vertex. $\newline$ For example, if $x=1$ : $\newline$ $y = 3(1)^2 - 7$ $\newline$ $y = 3 - 7$ $\newline$ $y = -4$ $\newline$ Plot the point $(1, -4)$ on the graph.
Find Point with Negative $x$ : Find another point using a negative value for $x$ . If $x=-1$ : $y = 3(-1)^2 - 7$ $y = 3 - 7$ $y = -4$ Plot the point $(-1, -4)$ on the graph. Notice that the point is symmetrical to the point $(1, -4)$ with respect to the vertex.
Draw Parabola: Draw the parabola. Using the vertex and the points found, draw a smooth curve to represent the parabola. Make sure the parabola opens upwards and is symmetrical about the $y$ -axis.

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