To analyze the quadratic function \( f(x) = 4 - (x - 3)^2 \), let's find the vertex, intercepts, axis of symmetry, domain, and range.
1. Vertex
The function is in vertex form \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Here, we have:
- \( h = 3 \)
- \( k = 4 \)
Thus, the vertex is at the point \((3, 4)\).
2. Intercepts
-
y-intercept: This is found by evaluating \( f(0) \): \[ f(0) = 4 - (0 - 3)^2 = 4 - 9 = -5 \] So the y-intercept is \( (0, -5) \).
-
x-intercepts: Set \( f(x) = 0 \): \[ 0 = 4 - (x - 3)^2 \] Rearranging gives: \[ (x - 3)^2 = 4 \] Taking the square root of both sides: \[ x - 3 = 2 \quad \text{or} \quad x - 3 = -2 \] Thus, \[ x = 5 \quad \text{or} \quad x = 1 \] So the x-intercepts are \( (1, 0) \) and \( (5, 0) \).
3. Axis of Symmetry
The axis of symmetry is the vertical line that passes through the vertex: \[ x = 3 \]
4. Domain
For quadratic functions, the domain is all real numbers: \[ \text{Domain: } (-\infty, \infty) \]
5. Range
Since the parabola opens downwards (the coefficient of \( (x-3)^2 \) is negative), the range will be all values less than or equal to the y-coordinate of the vertex: \[ \text{Range: } (-\infty, 4] \]
Summary
- Axis of symmetry: \( x = 3 \)
- Domain: \( (-\infty, \infty) \)
- Range: \( (-\infty, 4] \)
Now you can sketch the parabola using the vertex (3, 4) and the intercepts (0, -5), (1, 0), and (5, 0). The graph will be a downward-opening parabola.
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