Question
Solve the equation of a graph with a line of symmetry at x = 4, a range of (-∞, 9] and x -intercepts 3 units away from the line of symmetry
Answers
GPT-4o mini
To solve for the equation of a graph that meets the given conditions, let's break down the components:
1. **Line of Symmetry at \(x = 4\)**: This indicates that the graph is symmetric around the vertical line \(x = 4\). This is crucial for the form of the equation, which can generally be a quadratic expression or another type of function that has a vertical line of symmetry.
2. **Range of \((-∞, 9]\)**: This means that the maximum value of the graph is \(9\), and the graph approaches negative infinity as \(x\) moves away from this maximum. This is typically characteristic of a downward-opening parabola.
3. **\(x\)-Intercepts 3 units away from the line of symmetry**: The x-intercepts are located at positions \(3\) units away from the line of symmetry at \(x = 4\). Therefore, the \(x\)-intercepts can be calculated as follows:
- One x-intercept will be at \(4 - 3 = 1\).
- The other x-intercept will be at \(4 + 3 = 7\).
This gives us the x-intercepts at \(x = 1\) and \(x = 7\).
Now, we can express the equation of a quadratic function, which opens downward, using the vertex form:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex. For our graph, the vertex must be at \( (4, 9) \) since the highest point is \(9\) with symmetry at \(x = 4\). Thus, \(h = 4\) and \(k = 9\).
The equation becomes:
\[
y = a(x - 4)^2 + 9
\]
Next, we use the fact that the x-intercepts occur at \(x = 1\) and \(x = 7\). These values represent where \(y = 0\).
Let’s find the value of \(a\) using one of the intercepts, say \(x = 1\):
\[
0 = a(1 - 4)^2 + 9
\]
Calculating:
\[
0 = a(3)^2 + 9
\]
\[
0 = 9a + 9
\]
To solve for \(a\):
\[
9a = -9
\]
\[
a = -1
\]
Now, substituting \(a\) back into the equation gives us:
\[
y = -1(x - 4)^2 + 9
\]
Finally, we can expand this equation to standard form:
\[
y = -1(x^2 - 8x + 16) + 9
\]
\[
y = -x^2 + 8x - 16 + 9
\]
\[
y = -x^2 + 8x - 7
\]
Thus, the equation of the graph that meets all the specified criteria is
\[
y = -x^2 + 8x - 7.
\]
1. **Line of Symmetry at \(x = 4\)**: This indicates that the graph is symmetric around the vertical line \(x = 4\). This is crucial for the form of the equation, which can generally be a quadratic expression or another type of function that has a vertical line of symmetry.
2. **Range of \((-∞, 9]\)**: This means that the maximum value of the graph is \(9\), and the graph approaches negative infinity as \(x\) moves away from this maximum. This is typically characteristic of a downward-opening parabola.
3. **\(x\)-Intercepts 3 units away from the line of symmetry**: The x-intercepts are located at positions \(3\) units away from the line of symmetry at \(x = 4\). Therefore, the \(x\)-intercepts can be calculated as follows:
- One x-intercept will be at \(4 - 3 = 1\).
- The other x-intercept will be at \(4 + 3 = 7\).
This gives us the x-intercepts at \(x = 1\) and \(x = 7\).
Now, we can express the equation of a quadratic function, which opens downward, using the vertex form:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex. For our graph, the vertex must be at \( (4, 9) \) since the highest point is \(9\) with symmetry at \(x = 4\). Thus, \(h = 4\) and \(k = 9\).
The equation becomes:
\[
y = a(x - 4)^2 + 9
\]
Next, we use the fact that the x-intercepts occur at \(x = 1\) and \(x = 7\). These values represent where \(y = 0\).
Let’s find the value of \(a\) using one of the intercepts, say \(x = 1\):
\[
0 = a(1 - 4)^2 + 9
\]
Calculating:
\[
0 = a(3)^2 + 9
\]
\[
0 = 9a + 9
\]
To solve for \(a\):
\[
9a = -9
\]
\[
a = -1
\]
Now, substituting \(a\) back into the equation gives us:
\[
y = -1(x - 4)^2 + 9
\]
Finally, we can expand this equation to standard form:
\[
y = -1(x^2 - 8x + 16) + 9
\]
\[
y = -x^2 + 8x - 16 + 9
\]
\[
y = -x^2 + 8x - 7
\]
Thus, the equation of the graph that meets all the specified criteria is
\[
y = -x^2 + 8x - 7.
\]