How to convert factored form to vertex form?
Understand the Problem
The question is asking how to change a quadratic equation from its factored form into its vertex form. This process involves using algebraic methods to manipulate the equation and identify the vertex coordinates.
Answer
The vertex form of a quadratic equation is $y = a(x - h)^2 + k$, where $(h, k)$ represents the vertex coordinates.
Answer for screen readers
The vertex form of the quadratic equation can be expressed as $y = a(x - h)^2 + k$, where $(h, k)$ are the vertex coordinates derived from the original factored form.
Steps to Solve
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Identify the factored form of the quadratic equation The factored form typically looks like $y = a(x - p)(x - q)$, where $p$ and $q$ are the roots of the equation.
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Expand the factored form Use the distributive property (also known as FOIL) to expand the equation.
For example, if your equation is: $$ y = a(x - p)(x - q) $$ It expands to: $$ y = a(x^2 - (p+q)x + pq) $$
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Convert to standard form Write the equation in standard form $y = ax^2 + bx + c$. Here, $b = -a(p + q)$ and $c = a \cdot p \cdot q$.
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Find the vertex using the vertex formula The vertex can be found using the formula: $$ x = -\frac{b}{2a} $$ Substitute $b$ from the previous step to find the $x$-coordinate of the vertex.
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Calculate the $y$-coordinate of the vertex Substitute the $x$ value from the previous step back into the standard equation to find the $y$-coordinate.
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Write the vertex form of the quadratic The vertex form of a quadratic equation is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. Substitute the vertex coordinates into this form.
The vertex form of the quadratic equation can be expressed as $y = a(x - h)^2 + k$, where $(h, k)$ are the vertex coordinates derived from the original factored form.
More Information
Converting a quadratic from factored form to vertex form is useful in graphing the function, as it allows for a clear understanding of the vertex and the shape of the parabola. The vertex represents the minimum or maximum point of the quadratic function, depending on the value of $a$.
Tips
- Failing to expand the equation correctly when moving from factored form to standard form.
- Incorrectly calculating the vertex coordinates by not plugging the correct $x$ value back into the equation.
- Confusing the order of operations when evaluating the quadratic function for the vertex's $y$-coordinate.
