k(x)=−x2+6x−8k(x)=−x2+6x−8

Step 3 of 4: 

Find two points on the graph of the parabola other than the vertex and x-intercepts.




To find two points on the graph of the parabola represented by the function \( k(x) = -x^2 + 6x - 8 \) other than the vertex and \( x \)-intercepts, we can choose any \( x \)-values and compute the corresponding \( y \)-values using the function \( k(x) \).

### Step-by-Step Solution:

1. **Choose two \( x \)-values**:
   - We can choose any two \( x \)-values that are not the \( x \)-intercepts or the \( x \)-coordinate of the vertex. To avoid those, let's first find the vertex and the \( x \)-intercepts.

2. **Find the vertex**:
   The vertex of a parabola given by \( ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \).

   For \( k(x) = -x^2 + 6x - 8 \):
   - \( a = -1 \)
   - \( b = 6 \)

   So, the \( x \)-coordinate of the vertex is:
   \[
   x = -\frac{6}{2(-1)} = \frac{6}{2} = 3
   \]

   The \( x \)-coordinate of the vertex is \( x = 3 \).

3. **Find the \( x \)-intercepts**:
   The \( x \)-intercepts occur where \( k(x) = 0 \). To find them, solve the equation:
   \[
   -x^2 + 6x - 8 = 0
   \]

   Rearrange and factor the quadratic:
   \[
   x^2 - 6x + 8 = 0
   \]
   \[
   (x - 2)(x - 4) = 0
   \]

   So, the \( x \)-intercepts are at \( x = 2 \) and \( x = 4 \).

4. **Choose two other \( x \)-values**:
   We need to choose two \( x \)-values different from 2, 3, and 4. Let's choose \( x = 0 \) and \( x = 1 \).

5. **Compute the corresponding \( y \)-values**:
   - For \( x = 0 \):
     \[
     k(0) = -0^2 + 6(0) - 8 = -8
     \]
     So, one point on the graph is \( (0, -8) \).

   - For \( x = 1 \):
     \[
     k(1) = -(1)^2 + 6(1) - 8 = -1 + 6 - 8 = -3
     \]
     So, another point on the graph is \( (1, -3) \).

### Conclusion:

The two points on the graph of the parabola \( k(x) = -x^2 + 6x - 8 \), other than the vertex and the \( x \)-intercepts, are:
- \( (0, -8) \)
- \( (1, -3) \)

Would you like further clarification, or do you have any additional questions?

### Related Questions:

1. How do you find the \( y \)-intercept of a parabola?
2. What is the axis of symmetry for the parabola given by \( k(x) = -x^2 + 6x - 8 \)?
3. How do you determine whether a parabola opens upwards or downwards?
4. How would you complete the square to find the vertex form of \( k(x) \)?
5. What is the significance of the discriminant in the context of finding the \( x \)-intercepts?

### Tip:

When working with parabolas, knowing the vertex and the direction it opens helps in quickly sketching its graph. The \( x \)-intercepts provide additional points of intersection with the x-axis, making it easier to visualize.

Explore how to find points on the graph of the quadratic function k(x) = -x^2 + 6x - 8, excluding the vertex and x-intercepts. Follow a detailed step-by-step solution to determine specific points using different x-values. Ideal for students in Grades 9-10.

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