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 to find the equation of a parabola in vertex form given its vertex and a point it passes through. The vertex is (1, 0) and it passes through the point (-3, -4/3). The solution will involve substituting these values into the vertex form of a quadratic equation and solving for any necessary parameters.
Answer
$$ y = -\frac{1}{12}(x - 1)^2 $$
Answer for screen readers
The equation of the parabola in vertex form is: $$ y = -\frac{1}{12}(x - 1)^2 $$
Steps to Solve
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Identify the vertex and point The vertex of the parabola is given as $(1, 0)$. The point it passes through is $(-3, -\frac{4}{3})$.
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Write the vertex form of the equation The vertex form of a parabola is given by the equation: $$ y = a(x - h)^2 + k $$ where $(h, k)$ is the vertex. Here, $h = 1$ and $k = 0$. Thus, we can write: $$ y = a(x - 1)^2 + 0 $$ or simply: $$ y = a(x - 1)^2 $$
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Substitute the point into the vertex form Now, we will substitute the point $(-3, -\frac{4}{3})$ into the equation to find the value of $a$: $$ -\frac{4}{3} = a(-3 - 1)^2 $$
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Calculate the square First, calculate $(-3 - 1)$: $$ -3 - 1 = -4 $$ Then, calculate $(-4)^2$: $$ (-4)^2 = 16 $$
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Substitute back to find $a$ Now substitute back into the equation: $$ -\frac{4}{3} = a(16) $$ This simplifies to: $$ a = -\frac{4}{3 \times 16} $$ $$ a = -\frac{4}{48} = -\frac{1}{12} $$
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Write the final equation Now, we can substitute the value of $a$ back into the vertex form: $$ y = -\frac{1}{12}(x - 1)^2 $$
The equation of the parabola in vertex form is: $$ y = -\frac{1}{12}(x - 1)^2 $$
More Information
This vertex form of the equation is helpful because it allows you to directly see the vertex of the parabola and the effect of the coefficient $a$ on the width and direction of the parabola (in this case, it opens downward).
Tips
- Miscalculating the square of the step: Ensure you correctly compute expressions like $(-3 - 1)^2$.
- Incorrectly substituting values: Double-check that you substitute the correct point values into the equation.
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