A parabola opening up or down has vertex (1, 0) and passes through (-3, -4/3). Write its equation in vertex form. Simplify any fractions.
Understand the Problem
The question is asking for the equation of a parabola in vertex form, given the vertex (1, 0) and a point it passes through, (-3, -4/3). The process involves using the vertex form of a parabola and substituting the known values to derive the equation.
Answer
The equation of the parabola is given by $$ y = -\frac{1}{12}(x - 1)^2 $$
Answer for screen readers
The equation of the parabola in vertex form is
$$ y = -\frac{1}{12}(x - 1)^2 $$
Steps to Solve
- Identify 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.
- Substitute the vertex values into the equation
The vertex is given as ((1, 0)). Substituting (h = 1) and (k = 0) into the equation gives:
$$ y = a(x - 1)^2 + 0 $$
or simply:
$$ y = a(x - 1)^2 $$
- Use the point to find 'a'
The parabola passes through the point ((-3, -\frac{4}{3})). Substitute (x = -3) and (y = -\frac{4}{3}) into the equation:
$$ -\frac{4}{3} = a(-3 - 1)^2 $$
This simplifies to:
$$ -\frac{4}{3} = a(-4)^2 $$
Thus,
$$ -\frac{4}{3} = 16a $$
- Solve for 'a'
Now, solve for (a):
$$ a = -\frac{4}{3} \div 16 = -\frac{4}{48} = -\frac{1}{12} $$
- Write the final equation
Now substitute (a) back into the equation:
$$ y = -\frac{1}{12}(x - 1)^2 $$
This is the equation of the parabola in vertex form.
The equation of the parabola in vertex form is
$$ y = -\frac{1}{12}(x - 1)^2 $$
More Information
This form emphasizes the vertex ((1, 0)) and reveals the direction of the parabola (opening downwards due to the negative coefficient).
Tips
- Neglecting to substitute the point correctly: Make sure to plug in both (x) and (y) values correctly while solving for (a).
- Mistakes in solving for (a): Be careful with division; it's easy to miscalculate when simplifying fractions.
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