Write the function in vertex form.
y=-x2+3x+4

To write the given quadratic equation \( y = -x^2 + 3x + 4 \) in **vertex form**, we need to rewrite it in the form:

\[
y = a(x - h)^2 + k
\]

where \( (h, k) \) is the vertex of the parabola.

Let's follow the steps to complete the square:

### Step 1: Factor out the coefficient of \( x^2 \) from the first two terms.
The given equation is:
\[
y = -x^2 + 3x + 4
\]
Factor out the negative sign from the first two terms:
\[
y = -\left(x^2 - 3x\right) + 4
\]

### Step 2: Complete the square inside the parentheses.
To complete the square, take half of the coefficient of \( x \) (which is -3), square it, and add and subtract that value inside the parentheses.
\[
\left(\frac{-3}{2}\right)^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}
\]

Now add and subtract \( \frac{9}{4} \) inside the parentheses:
\[
y = -\left(x^2 - 3x + \frac{9}{4} - \frac{9}{4}\right) + 4
\]

### Step 3: Simplify the equation.
Now group the perfect square trinomial and simplify:
\[
y = -\left(\left(x - \frac{3}{2}\right)^2 - \frac{9}{4}\right) + 4
\]

Distribute the negative sign:
\[
y = -\left(x - \frac{3}{2}\right)^2 + \frac{9}{4} + 4
\]

Convert 4 into a fraction with the same denominator as \( \frac{9}{4} \):
\[
y = -\left(x - \frac{3}{2}\right)^2 + \frac{9}{4} + \frac{16}{4}
\]

Now add the fractions:
\[
y = -\left(x - \frac{3}{2}\right)^2 + \frac{25}{4}
\]

### Final Vertex Form:
The vertex form of the equation is:
\[
y = -\left(x - \frac{3}{2}\right)^2 + \frac{25}{4}
\]

Here, the vertex of the parabola is \( \left(\frac{3}{2}, \frac{25}{4}\right) \).

Would you like further details on the steps or have any questions?

### Related Questions:
1. How do you find the vertex of a quadratic equation from standard form?
2. What is the significance of the vertex in a parabola?
3. How does the sign of the leading coefficient affect the direction of the parabola?
4. How do you convert a quadratic equation in vertex form back to standard form?
5. Can all quadratic equations be converted to vertex form using completing the square?

### Tip:
When completing the square, always remember to balance your equation by both adding and subtracting the same value.

Write the function in vertex form. y=-x^2+3x+4

Learn how to convert the quadratic equation y=-x^2+3x+4 to vertex form using the completing the square method. This step-by-step guide covers essential algebraic concepts and formulas. Suitable for students in Grades 9-11.

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