https://file.aiquickdraw.com/mathbot/file/172722204040059net5cw.jpg

Let's go through each part of the question step by step.

### Part (a)
You are asked to express \( f(x) = x^2 - 4x + 5 \) in the form \( f(x) = (x - a)^2 + b \).

This requires completing the square:

1. Start with the quadratic \( f(x) = x^2 - 4x + 5 \).
2. To complete the square, take half of the coefficient of \( x \) (which is -4), square it, and add/subtract that inside the expression.

   \[
   f(x) = (x^2 - 4x + 4) + 5 - 4
   \]

3. This simplifies to:

   \[
   f(x) = (x - 2)^2 + 1
   \]

So, \( f(x) = (x - 2)^2 + 1 \), where \( a = 2 \) and \( b = 1 \).

### Part (b)
You need to sketch two graphs on the same diagram:
1. The graph of \( y = f(x) \), which is \( y = (x - 2)^2 + 1 \), a parabola shifted 2 units to the right and 1 unit upwards.
2. The graph of \( y = 10 - f(x) \), which becomes:

   \[
   y = 10 - \left( (x - 2)^2 + 1 \right) = 9 - (x - 2)^2
   \]

   This is an upside-down parabola shifted 2 units to the right and 9 units upwards.

### Part (c)
You need to find the range of values of \( x \) for which \( 10 - f(x) \) is positive. That is, solve:

\[
10 - f(x) > 0 \quad \text{or} \quad 10 > (x - 2)^2 + 1
\]

This simplifies to:

\[
9 > (x - 2)^2
\]

Taking the square root of both sides:

\[
-3 < x - 2 < 3
\]

So:

\[
-1 < x < 5
\]

Thus, the range of values for which \( 10 - f(x) \) is positive is \( x \in (-1, 5) \).

---

Would you like more details or have any questions?

### 5 Related Questions:
1. How do you complete the square for any quadratic expression?
2. What transformations occur when a parabola is shifted horizontally and vertically?
3. How can you determine the vertex of a quadratic function from its completed square form?
4. What happens graphically when you subtract a quadratic function from a constant?
5. How do you solve inequalities involving quadratic functions?

### Tip:
When completing the square, remember that the goal is to rewrite a quadratic function in a form that makes it easier to identify shifts and the vertex of the parabola.

Express f(x) = x^2 - 4x + 5 in the form f(x) = (x - a)^2 + b. On the same diagram sketch: the graph of y = f(x), the graph of y = 10 - f(x). Find the range of values of x for which 10 - f(x) is positive.

Explore how to express f(x) = x^2 - 4x + 5 in the form (x - a)^2 + b using completing the square method. Learn to graph y = f(x) and y = 10 - f(x) and solve quadratic inequalities. Perfect for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation