Question
Write the equation of the quadratic function in the graph in standard form. The axis of symmetry is at 3. -1.
The parabola has a y intercept at (0, 8) and a x intercept at (3, 0)
The parabola has a y intercept at (0, 8) and a x intercept at (3, 0)
Answers
GPT-4o mini
To write the quadratic function in standard form, we start with the information given.
1. **Axis of Symmetry**: The axis of symmetry is at \( x = 3 \). This means that the vertex form of the quadratic function can be started as \( y = a(x - 3)^2 + k \), where \( (3, k) \) is the vertex of the parabola.
2. **X-Intercept**: The x-intercept is at \( (3, 0) \). Since this point lies on the graph, we know that when \( x = 3 \), \( y = 0 \). This indicates that the vertex is also at \( (3, k) \) where \( k \) can be determined.
3. **Y-Intercept**: The y-intercept is at \( (0, 8) \). This means that when \( x = 0 \), \( y = 8 \). We can use this information to find the value of \( a \).
Since we have the x-intercept and the axis of symmetry, we can infer that the vertex of the parabola is also at \( (3, k) \).
The y-intercept gives us a point to plug into our equation once we find \( a \).
Given that the parabola opens upwards (assuming from the points provided):
1. We write the standard form based on the vertex and point given:
\[
y = a(x - 3)^2 + k
\]
2. Substituting the y-intercept point \( (0, 8) \) into the equation:
\[
8 = a(0 - 3)^2 + k
\]
\[
8 = 9a + k
\]
3. Since we also identified that the x-intercept is \( (3, 0) \), we can use the vertex point \( (3, k) \) to find \( k \):
When \( x = 3 \), \( y = 0 \):
\[
0 = a(3 - 3)^2 + k \implies k = 0
\]
4. Substituting \( k = 0 \) back into the equation from the y-intercept:
\[
8 = 9a + 0 \implies 9a = 8 \implies a = \frac{8}{9}
\]
Now substituting \( a \) and \( k \) into the vertex form equation gives:
\[
y = \frac{8}{9}(x - 3)^2
\]
Next, we need to convert this into standard form \( y = ax^2 + bx + c \).
1. Expanding the equation:
\[
y = \frac{8}{9}(x^2 - 6x + 9) = \frac{8}{9}x^2 - \frac{48}{9}x + \frac{72}{9}
\]
Simplifying the coefficients:
\[
y = \frac{8}{9}x^2 - \frac{16}{3}x + 8
\]
Thus, the equation of the quadratic function in standard form is:
\[
y = \frac{8}{9}x^2 - \frac{16}{3}x + 8
\]
1. **Axis of Symmetry**: The axis of symmetry is at \( x = 3 \). This means that the vertex form of the quadratic function can be started as \( y = a(x - 3)^2 + k \), where \( (3, k) \) is the vertex of the parabola.
2. **X-Intercept**: The x-intercept is at \( (3, 0) \). Since this point lies on the graph, we know that when \( x = 3 \), \( y = 0 \). This indicates that the vertex is also at \( (3, k) \) where \( k \) can be determined.
3. **Y-Intercept**: The y-intercept is at \( (0, 8) \). This means that when \( x = 0 \), \( y = 8 \). We can use this information to find the value of \( a \).
Since we have the x-intercept and the axis of symmetry, we can infer that the vertex of the parabola is also at \( (3, k) \).
The y-intercept gives us a point to plug into our equation once we find \( a \).
Given that the parabola opens upwards (assuming from the points provided):
1. We write the standard form based on the vertex and point given:
\[
y = a(x - 3)^2 + k
\]
2. Substituting the y-intercept point \( (0, 8) \) into the equation:
\[
8 = a(0 - 3)^2 + k
\]
\[
8 = 9a + k
\]
3. Since we also identified that the x-intercept is \( (3, 0) \), we can use the vertex point \( (3, k) \) to find \( k \):
When \( x = 3 \), \( y = 0 \):
\[
0 = a(3 - 3)^2 + k \implies k = 0
\]
4. Substituting \( k = 0 \) back into the equation from the y-intercept:
\[
8 = 9a + 0 \implies 9a = 8 \implies a = \frac{8}{9}
\]
Now substituting \( a \) and \( k \) into the vertex form equation gives:
\[
y = \frac{8}{9}(x - 3)^2
\]
Next, we need to convert this into standard form \( y = ax^2 + bx + c \).
1. Expanding the equation:
\[
y = \frac{8}{9}(x^2 - 6x + 9) = \frac{8}{9}x^2 - \frac{48}{9}x + \frac{72}{9}
\]
Simplifying the coefficients:
\[
y = \frac{8}{9}x^2 - \frac{16}{3}x + 8
\]
Thus, the equation of the quadratic function in standard form is:
\[
y = \frac{8}{9}x^2 - \frac{16}{3}x + 8
\]