To analyze and graph the quadratic function \( f(x) = (x - 10)^2 + 7 \), we can start by identifying key attributes: the vertex, intercepts, axis of symmetry, and the function's domain and range.
1. Finding the Vertex
The function is in the vertex form of a parabola, which is given by \( f(x) = a(x - h)^2 + k \). Here, the vertex \((h, k)\) can be determined directly from the function.
In this case:
- \( h = 10 \)
- \( k = 7 \)
So, the vertex is \((10, 7)\).
2. Finding the Axis of Symmetry
The axis of symmetry for a quadratic function in the form \( f(x) = a(x - h)^2 + k \) is given by the vertical line \( x = h \).
Thus, the axis of symmetry is: \[ x = 10 \]
3. Finding the Y-Intercept
To find the y-intercept, we set \( x = 0 \): \[ f(0) = (0 - 10)^2 + 7 = 100 + 7 = 107 \]
So, the y-intercept is \( (0, 107) \).
4. Finding the X-Intercepts
To find the x-intercepts, we set \( f(x) = 0 \): \[ (x - 10)^2 + 7 = 0 \] \[ (x - 10)^2 = -7 \]
This equation does not have real solutions since a square is never negative. This means the parabola does not intersect the x-axis.
5. Domain and Range
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Domain: Since this is a quadratic function, the domain is all real numbers: \[ \text{Domain: } (-\infty, \infty) \]
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Range: The vertex \( (10, 7) \) is the highest point of this parabola (opening upwards). Therefore, the minimum value of \( f(x) \) is \( 7 \): \[ \text{Range: } [7, \infty) \]
Summary
- The axis of symmetry is: \[ x = 10 \]
- The domain of \( f \) is: \[ (-\infty, \infty) \]
- The range of \( f \) is: \[ [7, \infty) \]
Graphing
You can plot the vertex (10, 7), the y-intercept (0, 107), and the axis of symmetry on a graph. Since the parabola opens upwards and does not touch the x-axis, it will be above the x-axis with its vertex at the peak.
To accurately sketch the graph, emphasize the shape of the parabola, starting from the vertex and confirming that it opens upward with the given properties stated above.