When determining the coordinates of the vertex for the function y = 2x² - 8x + 5, what are the coordinates?
Understand the Problem
The question is asking for the coordinates of the vertex of a quadratic function given in the standard form. To solve this, we can use the formula for the vertex of a parabola, which can be found at x = -b/(2a). Once we find x, we will substitute it back into the function to find the corresponding y coordinate.
Answer
The coordinates of the vertex are $(-\frac{b}{2a}, f(-\frac{b}{2a}))$.
Answer for screen readers
The coordinates of the vertex of the quadratic function are given by:
$$ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) $$
Steps to Solve
- Identify coefficients from the quadratic function
Let's denote the quadratic function in standard form as $f(x) = ax^2 + bx + c$. Identify the values of $a$, $b$, and $c$ from the given function.
- Calculate the x-coordinate of the vertex
Use the formula for the x-coordinate of the vertex, which is given by:
$$ x = -\frac{b}{2a} $$
Substitute the values of $b$ and $a$ to calculate $x$.
- Substitute x back into the function to find y
Now that we have the x-coordinate, substitute it back into the original function $f(x)$ to find the y-coordinate:
$$ y = f\left(-\frac{b}{2a}\right) $$
- Write the coordinates of the vertex
The coordinates of the vertex can now be expressed as a point, which we can write as:
$$(x, y)$$
where $x$ is what we computed in step 2 and $y$ is the result from step 3.
The coordinates of the vertex of the quadratic function are given by:
$$ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) $$
More Information
The vertex represents the maximum or minimum point of the parabola, depending on the sign of $a$. If $a > 0$, the parabola opens upwards and the vertex is a minimum point. If $a < 0$, the parabola opens downwards and the vertex is a maximum point.
Tips
- Confusing the signs of $a$ and $b$. Make sure to correctly identify whether they are positive or negative before applying the formula.
- Forgetting to substitute back into the original function to find the y-coordinate. It's essential to find both coordinates to express the vertex completely.
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