Question

Write the equation of the quadratic function in the graph in standard form. The axis of symmetry is at (-2, 5). The y intercept is at (0, 1). The x intercept is at (-4, 0).

y = _x^2 + _x + _

Answer 1

To write the equation of the quadratic function in standard form, we first need to understand that the general form of a quadratic function is:

\[ y = ax^2 + bx + c \]

Given the information:

1. The vertex (axis of symmetry) is at (-2, 5). This means the axis of symmetry is \( x = -2 \), and the vertex form of a quadratic function centered at the vertex \((h, k)\) is given by:

\[ y = a(x + 2)^2 + 5 \]

2. The y-intercept is at (0, 1). This means when \( x = 0 \), \( y = 1 \). We can use this to find \( a \).

3. The x-intercept is at (-4, 0). This point gives us another equation to work with.

First, we'll convert the vertex form to standard form using the y-intercept (0, 1):

\[
1 = a(0 + 2)^2 + 5
\]
\[
1 = a(2^2) + 5
\]
\[
1 = 4a + 5
\]
\[
4a = 1 - 5
\]
\[
4a = -4
\]
\[
a = -1
\]

Now that we know \( a = -1 \), we can write the equation in vertex form:

\[
y = -1(x + 2)^2 + 5
\]

Next, we expand this to get it into standard form:

\[
y = -1(x^2 + 4x + 4) + 5
\]
\[
y = -x^2 - 4x - 4 + 5
\]
\[
y = -x^2 - 4x + 1
\]

Thus, the equation of the quadratic function in standard form is:

\[
y = -1x^2 - 4x + 1
\]

So, the final values for the coefficients are:

- a = -1
- b = -4
- c = 1

Putting it all together:

\[
y = -x^2 - 4x + 1
\]