Question
What point is symmetrical to (−1,−3)
for the graph of y=x2+12x+8
?(1 point)
for the graph of y=x2+12x+8
?(1 point)
Answers
GPT-4o mini
To find the point that is symmetrical to the point \((-1, -3)\) with respect to the graph of the function \(y = x^2 + 12x + 8\), you will first need to find the vertex of the parabola, as symmetry in a parabola occurs around its vertex.
1. **Identify the equation of the parabola**: The equation is given as \(y = x^2 + 12x + 8\).
2. **Find the vertex**: The vertex form of a parabola given by \(y = ax^2 + bx + c\) can be found using the formula for the x-coordinate of the vertex, \(x = -\frac{b}{2a}\).
- Here, \(a = 1\) and \(b = 12\).
- The x-coordinate of the vertex is:
\[
x = -\frac{12}{2 \cdot 1} = -6
\]
- Now, substitute \(x = -6\) into the equation to find the y-coordinate:
\[
y = (-6)^2 + 12(-6) + 8 = 36 - 72 + 8 = -28
\]
- Therefore, the vertex of the parabola is \((-6, -28)\).
3. **Finding the symmetrical point**: The point symmetrical to \((-1, -3)\) with respect to the vertex \((-6, -28)\) can be found by reflecting it across the vertex.
- The x-coordinates are located at \(-1\) and \(-6\). The distance from \(-1\) to \(-6\) is \( |-1 - (-6)| = 5\). To find the symmetrical point, move 5 units left from the vertex, which leads us to:
\[
-6 - 5 = -11
\]
- Next, find the y-coordinate of the symmetrical point. Since we are reflecting vertically across the vertex:
- The y-coordinate distance from \(-3\) to \(-28\) is \( |-3 - (-28)| = 25\). Move down 25 units from the vertex:
\[
-28 + 25 = -3
\]
Therefore, the symmetrical point is:
\[
(-11, -3).
\]
In conclusion, the point symmetrical to \((-1, -3)\) for the graph of \(y = x^2 + 12x + 8\) is \(\boxed{(-11, -3)}\).
1. **Identify the equation of the parabola**: The equation is given as \(y = x^2 + 12x + 8\).
2. **Find the vertex**: The vertex form of a parabola given by \(y = ax^2 + bx + c\) can be found using the formula for the x-coordinate of the vertex, \(x = -\frac{b}{2a}\).
- Here, \(a = 1\) and \(b = 12\).
- The x-coordinate of the vertex is:
\[
x = -\frac{12}{2 \cdot 1} = -6
\]
- Now, substitute \(x = -6\) into the equation to find the y-coordinate:
\[
y = (-6)^2 + 12(-6) + 8 = 36 - 72 + 8 = -28
\]
- Therefore, the vertex of the parabola is \((-6, -28)\).
3. **Finding the symmetrical point**: The point symmetrical to \((-1, -3)\) with respect to the vertex \((-6, -28)\) can be found by reflecting it across the vertex.
- The x-coordinates are located at \(-1\) and \(-6\). The distance from \(-1\) to \(-6\) is \( |-1 - (-6)| = 5\). To find the symmetrical point, move 5 units left from the vertex, which leads us to:
\[
-6 - 5 = -11
\]
- Next, find the y-coordinate of the symmetrical point. Since we are reflecting vertically across the vertex:
- The y-coordinate distance from \(-3\) to \(-28\) is \( |-3 - (-28)| = 25\). Move down 25 units from the vertex:
\[
-28 + 25 = -3
\]
Therefore, the symmetrical point is:
\[
(-11, -3).
\]
In conclusion, the point symmetrical to \((-1, -3)\) for the graph of \(y = x^2 + 12x + 8\) is \(\boxed{(-11, -3)}\).