1. Problem: Express $2x^2 - 8x + 1$ in the form $a(x+b)^2 + c$ where $a$ and $b$ are integers. Step 1: Factor out the coefficient of $x^2$ from the first two terms: $$2x^2 - 8x + 1 = 2(x^2 - 4x) + 1$$ Step 2: Complete the square inside the parentheses: Take half of $-4$, which is $-2$, and square it: $(-2)^2 = 4$. Add and subtract 4 inside the parentheses: $$2(x^2 - 4x + 4 - 4) + 1 = 2((x - 2)^2 - 4) + 1$$ Step 3: Distribute 2 and simplify: $$2(x - 2)^2 - 8 + 1 = 2(x - 2)^2 - 7$$ Answer: $2x^2 - 8x + 1 = 2(x - 2)^2 - 7$ 2. Problem: Find the coordinates of the stationary point of $y = f(x) = 2x^2 - 8x + 1$. Step 1: Find the derivative: $$f'(x) = 4x - 8$$ Step 2: Set derivative to zero to find stationary points: $$4x - 8 = 0 \implies x = 2$$ Step 3: Find $y$ coordinate by substituting $x=2$ into $f(x)$: $$f(2) = 2(2)^2 - 8(2) + 1 = 8 - 16 + 1 = -7$$ Answer: Stationary point is at $(2, -7)$ 3. Problem: For $f(x) = x^2 - x - 5$, find the minimum value and corresponding $x$. Step 1: Derivative: $$f'(x) = 2x - 1$$ Step 2: Set derivative to zero: $$2x - 1 = 0 \implies x = \frac{1}{2}$$ Step 3: Find minimum value: $$f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - \frac{1}{2} - 5 = \frac{1}{4} - \frac{1}{2} - 5 = -\frac{9}{4} = -2.25$$ Answer: Minimum value is $-\frac{9}{4}$ at $x=\frac{1}{2}$ 4. Problem: Write a suitable domain for $f(x)$ so that $f^{-1}(x)$ exists. Step 1: Since $f(x)$ is a parabola opening upwards, it is not one-to-one on $\mathbb{R}$. Step 2: Restrict domain to $[\frac{1}{2}, \infty)$ where $f$ is increasing. Answer: Suitable domain is $x \geq \frac{1}{2}$ 5. Problem: For $f(x) = 5 - 7x - 2x^2$, write in the form $p - 2(x - q)^2$. Step 1: Rewrite: $$f(x) = -2x^2 - 7x + 5$$ Step 2: Factor out $-2$ from the $x$ terms: $$f(x) = -2\left(x^2 + \frac{7}{2}x\right) + 5$$ Step 3: Complete the square inside parentheses: Half of $\frac{7}{2}$ is $\frac{7}{4}$, square it: $\left(\frac{7}{4}\right)^2 = \frac{49}{16}$ Add and subtract inside: $$-2\left(x^2 + \frac{7}{2}x + \frac{49}{16} - \frac{49}{16}\right) + 5 = -2\left(\left(x + \frac{7}{4}\right)^2 - \frac{49}{16}\right) + 5$$ Step 4: Distribute $-2$: $$-2\left(x + \frac{7}{4}\right)^2 + \frac{98}{16} + 5 = -2\left(x + \frac{7}{4}\right)^2 + \frac{49}{8} + 5$$ Step 5: Convert 5 to eighths: $$5 = \frac{40}{8}$$ Sum constants: $$\frac{49}{8} + \frac{40}{8} = \frac{89}{8}$$ Answer: $$f(x) = \frac{89}{8} - 2\left(x + \frac{7}{4}\right)^2$$ 6. Problem: Write down the range of $f(x)$. Step 1: Since coefficient of squared term is negative, parabola opens downward. Step 2: Maximum value is $p = \frac{89}{8}$. Step 3: Range is all values less than or equal to $\frac{89}{8}$. Answer: Range is $(-\infty, \frac{89}{8}]$