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The question asks about the equation \( y = x^2 + 4x + 4 \), and a graph of \( y \) versus \( x \) is shown, along with points on the graph.

### Part (a): Express \( Y \) in terms of \( x \)
The given equation is already expressed as \( y = x^2 + 4x + 4 \). This is a quadratic equation, and it can be rewritten for further simplification. The equation can be factored as:
\[
y = (x + 2)^2
\]
Thus, the expression for \( Y \) in terms of \( x \) is:
\[
Y = (x + 2)^2
\]

### Part (b): Find the value of \( h \)
The graph shows a point labeled \( (h, 5) \), meaning that when \( y = 5 \), \( x = h \). We substitute \( y = 5 \) into the equation \( y = x^2 + 4x + 4 \):
\[
5 = x^2 + 4x + 4
\]
This simplifies to:
\[
x^2 + 4x - 1 = 0
\]
To solve for \( x \), we use the quadratic formula:
\[
x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-1)}}{2(1)}
\]
\[
x = \frac{-4 \pm \sqrt{16 + 4}}{2}
\]
\[
x = \frac{-4 \pm \sqrt{20}}{2}
\]
\[
x = \frac{-4 \pm 2\sqrt{5}}{2}
\]
\[
x = -2 \pm \sqrt{5}
\]
Therefore, \( h = -2 + \sqrt{5} \), as the graph shows the positive root for \( h \).

Would you like any further details?

Here are some additional questions related to this problem:
1. How would the graph change if the equation were \( y = x^2 + 4x + 1 \)?
2. What are the roots of the equation \( y = x^2 + 4x + 4 \)?
3. Can you derive the vertex of the quadratic function \( y = x^2 + 4x + 4 \)?
4. How would you describe the concavity of the graph of \( y = x^2 + 4x + 4 \)?
5. What is the significance of the discriminant in determining the value of \( h \)?

**Tip:** When solving quadratic equations, always check the discriminant (\( b^2 - 4ac \)) to determine if the solutions are real, repeated, or complex.

Two variables, x and y, are related by the equation y = x^2 + 4x + 4. Diagram 2 shows a straight line graph obtained by plotting Y against x. (a) Express Y in terms of x. (b) Hence, find the value of h.

Learn how to express Y in terms of x from the quadratic equation y = x^2 + 4x + 4, and find the value of h by solving the quadratic equation. The problem involves algebraic manipulation and the application of the quadratic formula. Suitable for students in Grades 10-11.

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