k(x)=(x−2)2−7k(x)=(x−2)2−7

Step 4 of 4 :  

Graph the parabola.




To graph the parabola for the function \( k(x) = (x - 2)^2 - 7 \), follow these steps:

### 1. **Identify the Vertex**
The function is in the form \( k(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

- Here, \( h = 2 \) and \( k = -7 \), so the vertex is \( (2, -7) \).

### 2. **Determine the Axis of Symmetry**
The axis of symmetry is a vertical line that passes through the vertex. For this function, the axis of symmetry is the line \( x = 2 \).

### 3. **Find the X-Intercepts**
We previously found the x-intercepts by solving \( k(x) = 0 \):
\[
(2 + \sqrt{7}, 0) \quad \text{and} \quad (2 - \sqrt{7}, 0)
\]
These are approximate points at \( x \approx 4.65 \) and \( x \approx -0.65 \).

### 4. **Find the Y-Intercept**
Set \( x = 0 \) to find the y-intercept:
\[
k(0) = (0 - 2)^2 - 7 = 4 - 7 = -3
\]
So the y-intercept is \( (0, -3) \).

### 5. **Plot the Points**
- Plot the vertex \( (2, -7) \).
- Plot the x-intercepts \( (2 + \sqrt{7}, 0) \) and \( (2 - \sqrt{7}, 0) \).
- Plot the y-intercept \( (0, -3) \).
- Note the symmetry of the parabola around the line \( x = 2 \).

### 6. **Draw the Parabola**
- Sketch a smooth curve through the plotted points, opening upwards (since the coefficient of the squared term is positive).
- The curve should pass through the vertex, x-intercepts, and y-intercept.

### Visualization:
To visualize the graph, here's a rough sketch of the parabola:

- The vertex \( (2, -7) \) is the lowest point.
- The parabola opens upwards.
- It crosses the x-axis at \( (2 + \sqrt{7}, 0) \) and \( (2 - \sqrt{7}, 0) \).
- It crosses the y-axis at \( (0, -3) \).

If you need a precise graph, graphing software or plotting on graph paper would be useful.

Would you like more details or have any questions? Here are some related questions:

1. How does the width of this parabola compare to \( k(x) = x^2 \)?
2. What happens to the graph if the equation is \( k(x) = -(x - 2)^2 - 7 \)?
3. How do you find the focus and directrix of the parabola \( k(x) = (x - 2)^2 - 7 \)?
4. Can you determine the range of the function \( k(x) = (x - 2)^2 - 7 \)?
5. How does shifting the vertex affect the graph of the parabola?

**Tip:** When sketching parabolas, always plot the vertex and a few key points, then use the axis of symmetry to help guide your sketch.

Learn how to graph the parabola for the quadratic function k(x) = (x - 2)^2 - 7. Understand the vertex, axis of symmetry, x-intercepts, and y-intercept through a step-by-step solution. Suitable for high school students studying quadratic functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation