1. **State the problem:** Write each given function in the standard quadratic form $ax^2 + bx + c$ and identify the values of $a$, $b$, and $c$. 2. **Recall the standard form:** A quadratic function is written as $$f(x) = ax^2 + bx + c$$ where $a$, $b$, and $c$ are constants, and $a \neq 0$. 3. **Analyze each function:** - a. $f(x) = 2x - x^2 + 1$ Rearrange terms to standard form: $$f(x) = -x^2 + 2x + 1$$ So, $a = -1$, $b = 2$, $c = 1$. - b. $g(x) = 5 + 3x + x^2$ Rearrange terms: $$g(x) = x^2 + 3x + 5$$ So, $a = 1$, $b = 3$, $c = 5$. - c. $h(t) = t^2 - 2t + 1$ Already in standard form: $$h(t) = t^2 - 2t + 1$$ So, $a = 1$, $b = -2$, $c = 1$. - d. $f(x) = x^2 - 2x + 3$ Already in standard form: $$f(x) = x^2 - 2x + 3$$ So, $a = 1$, $b = -2$, $c = 3$. - e. $h(x) = 8x^2 + 4 - 4x$ Rearrange terms: $$h(x) = 8x^2 - 4x + 4$$ So, $a = 8$, $b = -4$, $c = 4$. 4. **Summary table:** | Function | Expression in $ax^2 + bx + c$ | $a$ | $b$ | $c$ | |---|---|---|---|---| | a | $-x^2 + 2x + 1$ | $-1$ | $2$ | $1$ | | b | $x^2 + 3x + 5$ | $1$ | $3$ | $5$ | | c | $t^2 - 2t + 1$ | $1$ | $-2$ | $1$ | | d | $x^2 - 2x + 3$ | $1$ | $-2$ | $3$ | | e | $8x^2 - 4x + 4$ | $8$ | $-4$ | $4$ | 5. **Check if each function is quadratic:** Since all have $a \neq 0$, all are quadratic functions (QF). **Final answer:** | Function | Quadratic? | |---|---| | a | QF | | b | QF | | c | QF | | d | QF | | e | QF |