A parabola opening up or down has vertex (0, 0) and passes through (-4, 4). Write its equation in vertex form. Simplify any fractions.
Understand the Problem
The question is asking to determine the equation of a parabola given its vertex at (0, 0) and another point it passes through, (-4, 4). The equation needs to be provided in vertex form.
Answer
The equation of the parabola is $$ y = \frac{1}{4}x^2 $$
Answer for screen readers
The equation of the parabola in vertex form is
$$ y = \frac{1}{4}x^2 $$
Steps to Solve
- Understand the vertex form of a parabola
The vertex form of a parabola is given by the equation:
$$ y = a(x - h)^2 + k $$
where $(h, k)$ is the vertex of the parabola. In this case, the vertex is at $(0, 0)$, so the equation simplifies to:
$$ y = ax^2 $$
- Substitute the known point into the equation
We know the parabola passes through the point $(-4, 4)$. We will substitute these values into the equation to solve for $a$:
$$ 4 = a(-4)^2 $$
This simplifies to:
$$ 4 = 16a $$
- Solve for coefficient (a)
Now, isolate (a):
$$ a = \frac{4}{16} $$
Simplifying gives:
$$ a = \frac{1}{4} $$
- Write the final equation in vertex form
Now that we have the value of (a), we can write the equation of the parabola in vertex form:
$$ y = \frac{1}{4}x^2 $$
The equation of the parabola in vertex form is
$$ y = \frac{1}{4}x^2 $$
More Information
In this case, the parabola opens upwards, and the value of (a) determines how "wide" or "narrow" the parabola is. A smaller absolute value of (a) means a wider parabola.
Tips
- Forgetting to square the x value when substituting for (a).
- Failing to simplify (a) correctly. Ensure to reduce fractions to their simplest form.
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