Question

Use the vertex and intercepts to sketch the graph of the
quadratic function. Give the equation of the parabola's
axis of symmetry. Use the graph to determine
the function's domain and range.

y-5=(x-1)^2

Use the graphing tool to graph the equation. Use the
vertex and one of the intercepts when drawing the
graph.


The axis of symmetry is blank
(Type an equation.)

The domain of the function is blank
(Type your answer in interval notation.)

The range of the function is blank
(Type your answer in interval notation.)

Answer 1

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
- **Domain**: The domain of any quadratic function is all real numbers.
\[
\text{Domain: } (-\infty, \infty)
\]

- **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.