Use the vertex and intercepts to sketch the graph of the quadratic function. Give the equation for the parabola's axis of symmetry. Use the parabola to identify the function's domain and range. \[ f(x)=16-(x-1)^{2} \] \( \square \) The axis of symmetry is \( \mathrm{x}=1 \). (Type an equation. Simplify your answer.) Identify the function's domain. The domain is \( (-\infty, \infty) \). (Type the answer in interval notation.) Identify the function's range. The range is \( \square \) (Type the answer in iferval notation.)

To sketch the graph of the quadratic function \( f(x) = 16 - (x-1)^{2} \), we first note that the vertex is at the point \( (1, 16) \). Since the quadratic opens downwards (as indicated by the negative coefficient in front of the squared term), the y-intercept occurs when \( x = 0 \), yielding \( f(0) = 16 - (0-1)^{2} = 15 \). The x-intercepts can be found by setting \( f(x) = 0 \), resulting in \( 16 - (x-1)^{2} = 0 \), which leads to \( (x-1)^{2} = 16 \) and gives the intercepts \( x = 1 \pm 4 \), or \( x = -3 \) and \( x = 5 \). The parabola's axis of symmetry is given by the equation \( x = 1 \), and its domain is all real numbers, expressed in interval notation as \( (-\infty, \infty) \). The range is from the maximum value at the vertex down to negative infinity, so the range is \( (-\infty, 16] \). The axis of symmetry is \( \mathrm{x}=1 \). The domain is \( (-\infty, \infty) \). The range is \( (-\infty, 16] \). Did you know that parabolas can model a variety of real-world phenomena? Think of the path a basketball makes when shooting into a hoop—it follows a parabolic trajectory! This shape is vital in engineering, physics, and even architecture. Parabolic arches, for example, distribute weight efficiently and look stunning as bridges or grand entrances. If you're sketching parabolas, remember to check for sign errors in your calculations. A common mistake is miscalculating the vertex or intercepts, which can throw off the entire graph. To avoid this, double-check your work, particularly with the vertex formula \( x = -\frac{b}{2a} \) for standard form, and always remember to plot a few points on either side of your vertex to ensure a smooth curve!