Problem 25 - Beginning and Intermediate Algebra

Question

Graph each parabola. Give the vertex, axis of symmetry, domain, and range. $$ f(x)=x^{2}+8 x+10 $$

Step-by-Step Solution

Answer

Vertex: (0, 0), Axis of Symmetry: x = 0, Domain: (-\infty, \infty), Range: (-\infty, 0].

Step 1 - Identify the Vertex

The standard form of a parabola is given by \( f(x) = ax^2 + bx + c \). In this case, the given function is \( f(x) = -2x^2 \), which is in the format of \( ax^2 \). The vertex form is \( f(x) = a(x-h)^2 + k \). Here, \( h = 0 \) and \( k = 0 \), so the vertex is (0, 0).

Step 2 - Determine the Axis of Symmetry

The axis of symmetry for a parabola in the form \( ax^2 + bx + c \) is given by the line \( x = -\frac{b}{2a} \). Since \( b = 0 \) in our function \( f(x) = -2x^2 \), the axis of symmetry is \( x = 0 \).

Step 3 - Identify the Domain

The domain of any quadratic function is all real numbers. Thus, the domain of \( f(x) = -2x^2 \) is \( (-\infty, +\infty) \).

Step 4 - Identify the Range

For the range, observe that the parabola opens downwards because the coefficient of \( x^2 \) is negative. Thus, the maximum value is the y-coordinate of the vertex. Therefore, the range is \( (-\infty, 0] \).

Step 5 - Graph the Parabola

Plot the vertex at (0, 0). Since the leading coefficient is \( -2 \), the parabola opens downwards. Sketch the curve symmetrical about the y-axis, passing through the points consistent with \( y = -2x^2 \).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is the point where the curve changes direction. In the given problem, the function is written as \( f(x) = -2x^2 \). This is a simple form where the squared term is not shifted horizontally or vertically. Thus, both the horizontal (h) and vertical (k) shifts are zero. The general form of a parabola is \( f(x) = a(x-h)^2 + k \). Here, \( h \) and \( k \) set the vertex of the parabola. When \( h = 0 \) and \( k = 0 \), the vertex is at (0, 0). For our problem, the vertex is thus clearly at the origin. This point is crucial for graphing, as it helps anchor the parabola on the coordinate plane.

Axis of Symmetry

The axis of symmetry is a vertical line that runs through the vertex of the parabola. It divides the parabola into two mirror images. For the equation \( f(x) = -2x^2 \), the axis of symmetry can be found using the formula \( x = -\frac{b}{2a} \). In this case, the formula simplifies because \( b = 0 \). Therefore, the axis of symmetry is simply \( x = -\frac{0}{2(-2)} = 0 \). This means the vertical line passing through \( x = 0 \) (the y-axis) is our axis of symmetry. This line helps understand the symmetrical nature of the parabola, making graphing easier.

Domain and Range

The domain of a parabola refers to all possible x-values (\( x \)) for which the function is defined. For any quadratic equation like \( f(x) = -2x^2 \), the domain is all real numbers. This is because you can input any real number, and the function will output a corresponding y-value.

The range of a parabola deals with the y-values (\( y \)) the function can output. For \( f(x) = -2x^2 \), the parabola opens downwards since the coefficient of \( x^2 \) is negative. Therefore, the y-values get larger as the x-values approach 0, peaking at the vertex before decreasing towards negative infinity. The highest y-value occurs at the vertex, which is at (0,0). Hence, the range is from negative infinity to zero, inclusive. Formally, we can write the range as \( (-\infty, 0] \). Understanding the domain and range ensures you know the extent of the parabola on the coordinate plane.