# How do you graph the parabola #3x^2+18x+21# using vertex, intercepts and additional points?

See method and graph below.

We will manipulate this to find points on the graph, and thus we can sketch the parabola.

Start off with the y-intercept, and make clear everything that you are doing (anything formatted you should write down).

So we know there is a y-intercept of 21. Now, for the x-intercepts

A quick check of the discriminant tells us that this does not factorise (the discriminant is 8 which is not a square number). So to solve this, we might as well use the Quadratic formula.

We don't need to find the values of these as a decimal, since when we sketch our graph, we will give the x-intercepts as exact values. The decimals are (if you're interested) round about -4.41 and -1.59.

This is enough to sketch the graph, although the question asks us to find the vertex (turning point), so we should probably find that too. You can complete the square or differentiate to find the turning point.

Completing the square

Differentiation

If the question didn't specify, and I had real roots to my quadratic, I wouldn't bother to find the turning point. The x-coordinate of the turning point will always lie halfway between the roots.

Now we have enough information to sketch the curve. Make sure to mark all of your intercepts exactly (even though this graph leaves it as a decimal). Aim for a smooth curve.

When sketching, you don't need to have the y-axis go up in the same step as the x-axis. Just make sure you label your points. (If you need me to go into more detail on this, I will do, just ask).

graph{y=3x^2+18x+21 [-31.85, 33.1, -7.4, 25.07]}

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To graph the parabola 3x^2 + 18x + 21, first, find the vertex using the formula x = -b / (2a). Substitute a = 3 and b = 18 into the formula to get x = -3. Then, find the y-coordinate of the vertex by substituting x = -3 into the equation to get y = 0. Next, find the x-intercepts by setting y = 0 and solving the quadratic equation 3x^2 + 18x + 21 = 0. You can factor the equation or use the quadratic formula. The x-intercepts are (-3, 0) and (-7, 0). Finally, plot these points on the graph and find additional points by choosing x-values and substituting them into the equation to find corresponding y-values.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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