In Exercises 57–62, use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function?
$y=-x^2+4x-3$

Verified step by step guidance

1. Identify the given quadratic function: $y=\; -x^2+\; 4x-\; 3$.
2. Determine the direction in which the parabola opens by looking at the coefficient of $x^2$. Since it is negative (-1), the parabola opens downward.
3. Find the vertex of the parabola using the vertex formula for $x$: $x=\; -\backslash frac\{b\}\{2a\}$, where $a\; =\; -1$ and $b\; =\; 4$. Calculate $x\; =\; -\backslash frac\{4\}\{2(-1)\}$.
4. Substitute the $x$-value of the vertex back into the original equation to find the $y$-coordinate of the vertex, which gives the maximum value of $y$ because the parabola opens downward.
5. Determine the domain and range: The domain of any quadratic function is all real numbers, so $\backslash text\{Domain\}\; =\; (-\backslash infty,\; \backslash infty)$. The range is all $y$-values less than or equal to the vertex's $y$-coordinate, so $\backslash text\{Range\}\; =\; (-\backslash infty,\; y\_\{vertex\}]$. Since each $x$ corresponds to exactly one $y$, the relation is a function.

Key Concepts

Vertex of a Parabola

The vertex is the highest or lowest point on a parabola, found using the formula x = -b/(2a) for a quadratic y = ax^2 + bx + c. It helps identify the maximum or minimum value of the function, which is crucial for determining the range.

Direction of the Parabola

The direction a parabola opens depends on the coefficient 'a' in y = ax^2 + bx + c. If 'a' is positive, it opens upward; if negative, downward. This direction indicates whether the vertex is a maximum or minimum point, affecting the range of the relation.

Domain and Range of Quadratic Functions

The domain of a quadratic function is all real numbers since x can take any value. The range depends on the vertex and the parabola's direction: if it opens downward, the range is all y-values less than or equal to the vertex's y-coordinate; if upward, all y-values greater than or equal to it.