Question
y=3x^2-6x+9 the high or low of this parabola is...
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GPT-4o mini
To determine whether the parabola represented by the equation \( y = 3x^2 - 6x + 9 \) has a high or low point, we can analyze its structure.
This equation is a quadratic function of the form \( y = ax^2 + bx + c \), where:
- \( a = 3 \) (which is positive),
- \( b = -6 \),
- \( c = 9 \).
Since the coefficient \( a \) is positive, this means the parabola opens upwards. Therefore, the vertex of the parabola will be a minimum point (the lowest point).
To find the vertex, we can use the vertex formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1
\]
Now, substitute \( x = 1 \) back into the original equation to find the \( y \)-coordinate of the vertex:
\[
y = 3(1)^2 - 6(1) + 9
\]
\[
y = 3(1) - 6 + 9
\]
\[
y = 3 - 6 + 9 = 6
\]
Thus, the vertex of the parabola is at the point \( (1, 6) \), and since it opens upwards, this point represents the low point (minimum) of the parabola.
In summary, the parabola has a **low** point at \( (1, 6) \).
This equation is a quadratic function of the form \( y = ax^2 + bx + c \), where:
- \( a = 3 \) (which is positive),
- \( b = -6 \),
- \( c = 9 \).
Since the coefficient \( a \) is positive, this means the parabola opens upwards. Therefore, the vertex of the parabola will be a minimum point (the lowest point).
To find the vertex, we can use the vertex formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1
\]
Now, substitute \( x = 1 \) back into the original equation to find the \( y \)-coordinate of the vertex:
\[
y = 3(1)^2 - 6(1) + 9
\]
\[
y = 3(1) - 6 + 9
\]
\[
y = 3 - 6 + 9 = 6
\]
Thus, the vertex of the parabola is at the point \( (1, 6) \), and since it opens upwards, this point represents the low point (minimum) of the parabola.
In summary, the parabola has a **low** point at \( (1, 6) \).