To analyze the quadratic function given in the vertex form \( y - 5 = (x - 1)^2 \), we can start by identifying important features such as the vertex, intercepts, axis of symmetry, domain, and range.
1. Vertex
The equation is in the vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex.
From the equation:
- \(h = 1\)
- \(k = 5\)
So, the vertex is \((1, 5)\).
2. Axis of Symmetry
The axis of symmetry can be given by the equation \(x = h\). Thus, the axis of symmetry is: \[ x = 1 \]
3. Finding Intercepts
Y-Intercept:
To find the y-intercept, set \(x = 0\): \[ y - 5 = (0 - 1)^2 \] \[ y - 5 = 1 \implies y = 6 \] Thus, the y-intercept is \((0, 6)\).
X-Intercepts:
To find the x-intercepts, set \(y = 0\): \[ 0 - 5 = (x - 1)^2 \] \[ -5 = (x - 1)^2 \] Since a square cannot be negative, there are no real x-intercepts.
4. Domain and Range
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Domain: The domain of any quadratic function is all real numbers. \[ \text{Domain: } (-\infty, \infty) \]
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Range: Since the vertex is the highest point (5) and the parabola opens upwards (the coefficient of \((x - 1)^2\) which is 1 is positive), the range is: \[ \text{Range: } [5, \infty) \]
Summary
- The axis of symmetry is \( x = 1 \).
- The domain of the function is \( (-\infty, \infty) \).
- The range of the function is \( [5, \infty) \).
Graphing
To sketch the graph, plot the vertex \((1, 5)\) and the y-intercept \((0, 6)\). Draw a parabola that opens upwards, ensuring it is symmetric around the line \(x = 1\).
Feel free to use a graphing tool for a more precise sketch.
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