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 write the equation of a parabola in vertex form based on the given vertex and a point it passes through. We need to apply the vertex form of a quadratic equation and utilize the coordinates provided.
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
- Identify the vertex and a point on the parabola
The vertex is given as $(0, 0)$, and the parabola passes through the point $(-4, 4)$.
- Write the vertex form of the parabola's equation
The vertex form of a parabola is given by the equation:
$$ y = a(x - h)^2 + k $$
where $(h, k)$ is the vertex. For our case, this becomes:
$$ y = a(x - 0)^2 + 0 $$
or simply:
$$ y = ax^2 $$
- Substitute the known point into the equation
We will substitute the point $(-4, 4)$ into the equation to find the value of $a$:
$$ 4 = a(-4)^2 $$
Calculating $(-4)^2$ gives $16$, so we have:
$$ 4 = 16a $$
- Solve for $a$
To find $a$, divide both sides by $16$:
$$ a = \frac{4}{16} = \frac{1}{4} $$
- Write the final equation
Substituting $a$ back into the vertex form gives:
$$ y = \frac{1}{4}x^2 $$
The equation of the parabola in vertex form is:
$$ y = \frac{1}{4}x^2 $$
More Information
This represents a parabola that opens upwards, with its vertex located at the origin $(0, 0)$, and passes through the point $(-4, 4)$. The coefficient $\frac{1}{4}$ determines the width of the parabola.
Tips
- Confusing the vertex coordinates or using the wrong point to substitute in the equation.
- Forgetting to correctly square the x-value from the point when substituting it into the equation.
- Misinterpreting the vertex form and incorrectly rearranging or simplifying the equation.
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