Problem 79
Question
If you are given the standard form of the equation of a parabola with vertex at the origin, explain how to determine if the parabola opens to the right, left, upward, or downward.
Step-by-Step Solution
Verified Answer
The direction of a parabola in the vertex form is determined by the coefficient 'a' in the equation. If 'a' is positive, the parabola opens upward when the equation revolves around y or right when it's around x. If 'a' is negative, the parabola opens downward when around y or left when around x.
1Step 1: Identify the form of the equation
Firstly, identify whether the equation is in the format \(y=a(x-h)^2+k\) or \(x=a(y-k)^2+h\). The equation will look similar to one of these patterns. The key is to look at which variable, x or y, is alone on one side of the equation.
2Step 2: Identify the coefficient
After identifying the right format, identify the coefficient 'a' in the equation. The coefficient of the square term will tell you the direction in which the parabola opens.
3Step 3: Determine the direction
If 'a' is greater than 0 and y is alone on one side, the parabola opens upward. If 'a' is less than 0 and y is alone on one side, the parabola opens downward. If 'a' is greater than 0 and x is alone on one side, the parabola opens to the right. If 'a' is less than 0 and x is alone on one side, the parabola opens to the left.
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