283 Give the domain and range of the quadratic function whose graph is described.

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To determine the **domain** and **range** of a quadratic function, let's start by understanding the general form of a quadratic function and its graph. ### General Form of a Quadratic Function A quadratic function can be written as: $$ f(x) = ax^2 + bx + c $$ where: - $$ a $$, $$ b $$, and $$ c $$ are constants - $$ a \neq 0 $$ (if $$ a = 0 $$, the function is linear, not quadratic) ### Graph of a Quadratic Function The graph of a quadratic function is a **parabola**. The direction in which the parabola opens (upward or downward) depends on the coefficient $$ a $$: - If $$ a > 0 $$: The parabola opens **upward**. - If $$ a < 0 $$: The parabola opens **downward**. The vertex of the parabola represents the **minimum** point (if $$ a > 0 $$) or the **maximum** point (if $$ a < 0 $$) of the function. ### Domain of a Quadratic Function The **domain** of a function is the set of all possible input values (typically $$ x $$-values) for which the function is defined. - **Domain**: For any quadratic function, there are no restrictions on the input values. You can plug in any real number for $$ x $$. $$ \text{Domain}: \quad (-\infty, \infty) $$ ### Range of a Quadratic Function The **range** of a function is the set of all possible output values (typically $$ y $$-values) the function can produce. - **If $$ a > 0 $$** (parabola opens upward): - The vertex is the **minimum** point. - The range starts from the $$ y $$-coordinate of the vertex and extends to infinity. - **If $$ a < 0 $$** (parabola opens downward): - The vertex is the **maximum** point. - The range starts from negative infinity and extends up to the $$ y $$-coordinate of the vertex. To find the exact range, you need the coordinates of the vertex, which can be found using: $$ \text{Vertex } (h, k) $$ where: $$ h = -\frac{b}{2a} $$ $$ k = f(h) = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c $$ - **Range**: - If $$ a > 0 $$: $$ [k, \infty) $$ - If $$ a < 0 $$: $$ (-\infty, k] $$ ### Example Let's consider a specific example to illustrate: **Function**: $$ f(x) = 2x^2 - 4x + 1 $$ 1. **Determine the Direction**: - $$ a = 2 > 0 $$: Parabola opens **upward**. 2. **Find the Vertex**: $$ h = -\frac{b}{2a} = -\frac{-4}{2 \times 2} = 1 $$ $$ k = f(1) = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1 $$ - Vertex: $$ (1, -1) $$ 3. **Domain**: $$ (-\infty, \infty) $$ 4. **Range**: - Since the parabola opens upward, the minimum $$ y $$-value is $$ -1 $$. $$ [-1, \infty) $$ ### Summary - **Domain**: All real numbers, $$ (-\infty, \infty) $$ - **Range**: - If the parabola opens **upward** ($$ a > 0 $$): $$[k, \infty)$$ - If the parabola opens **downward** ($$ a < 0 $$): $$(-\infty, k]$$ Where $$ k $$ is the $$ y $$-coordinate of the vertex. **Domain**: All real numbers **Range**: - If the parabola opens upward ($$ a > 0 $$): $$ [k, \infty) $$ - If the parabola opens downward ($$ a < 0 $$): $$ (-\infty, k] $$

283 Give the domain and range of the quadratic function whose graph is described.

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