What is the equation in vertex form of a parabola with a vertex of (4, −2) that passes through (2, −14)?
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 form of a parabola is expressed as y = a(x - h)² + k, where (h, k) is the vertex. We will use the given vertex and the point to find the value of 'a'.
Answer
The equation of the parabola is $$ y = (x - 2)^2 + 3 $$
Answer for screen readers
The equation of the parabola in vertex form is
$$ y = (x - 2)^2 + 3 $$
Steps to Solve
- Identify the vertex and a point on the parabola
Let's denote the vertex as $(h, k)$ and the point the parabola passes through as $(x_1, y_1)$. For example, if the vertex is $(2, 3)$ and the point is $(4, 7)$, we have $h = 2$, $k = 3$, $x_1 = 4$, and $y_1 = 7$.
- Write the vertex form of the parabola
Now, we can plug the vertex values into the vertex form of the parabola, which is: $$ y = a(x - h)^2 + k $$ Substituting in our vertex coordinates, we have: $$ y = a(x - 2)^2 + 3 $$
- Substitute the point into the equation to find 'a'
Next, substitute the coordinates of the point $(x_1, y_1)$ into the equation we have: $$ 7 = a(4 - 2)^2 + 3 $$ This simplifies to: $$ 7 = a(2)^2 + 3 $$
- Solve for 'a'
Now we simplify and solve for 'a': $$ 7 = 4a + 3 $$ Subtract 3 from both sides: $$ 4 = 4a $$ Divide both sides by 4: $$ a = 1 $$
- Write the final equation of the parabola
Now substitute the value of 'a' back into the vertex form: $$ y = 1(x - 2)^2 + 3 $$ This simplifies to: $$ y = (x - 2)^2 + 3 $$
The equation of the parabola in vertex form is
$$ y = (x - 2)^2 + 3 $$
More Information
The vertex form of a parabola provides an easy way to identify the vertex and the direction in which it opens. In this case, the parabola opens upwards because the value of 'a' is positive.
Tips
- Forgetting to square the term $(x - h)$ when applying the vertex form equation.
- Incorrect arithmetic operations while solving for 'a'.
- Mislabeling points or vertices by mixing up their coordinates.
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