Problem 41 - Beginning and Intermediate Algebra

Question

Graph each parabola. Give the vertex, axis of symmetry, domain, and range. $$ f(x)=-2(x+3)^{2}+4 $$

Step-by-Step Solution

Answer

The vertex is (-3, 4), the axis of symmetry is x = -3, the domain is (-∞, ∞), and the range is (-∞, 4].

Step 1: Identify the Standard Form

The given function is in the standard form of a parabola, expressed as y = a(x-h)^2 + kwhere (h,k) is the vertex of the parabola. Here, a = -2, h = -3 and k = 4.

Step 2: Find the Vertex

Substitute the values of h and k into the vertex form. Therefore, the vertex is (-3, 4).

Step 3: Determine the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. This line has the equation x = -3.

Step 4: Identify the Domain

Since the function is a parabola, the domain is all real numbers. Therefore, the domain is (-∞, ∞).

Step 5: Determine the Range

Because the coefficient of y =-2(x+3)^2 + 4(a = -2) is negative, the parabola opens downwards. The maximum value of the parabola is 4 (the y-coordinate of the vertex), so the range is (-∞, 4].

Step 6: Graph the Parabola

Plot the vertex (-3, 4) and draw the axis of symmetry x = -3. Then sketch the parabola opening downward from the vertex, making sure to reflect points symmetrically about the axis of symmetry.

Key Concepts

vertex of a parabola

The vertex of a parabola is a crucial point that represents either the maximum or minimum value of the function, depending on the direction in which the parabola opens. For the given equation \( f(x) = -2(x+3)^2 + 4 \), the vertex form is expressed as \( y = a(x-h)^2 + k \). Here, (h, k) is the vertex.

Let's identify h and k from our equation:

• The value of h is -3 (note the negative sign within the square term).
• The value of k is 4.

So, the vertex of the parabola is at (-3, 4). This is the highest point of the parabola because the coefficient a (-2) is negative, indicating the parabola opens downwards.

Remember, the vertex is your starting point when graphing a parabola.

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 image halves.

For our equation \( f(x) = -2(x+3)^2 + 4 \), the axis of symmetry can be found directly from the vertex:

• The vertex is at (-3, 4).
• The axis of symmetry is the line x = -3.

By plotting this vertical line on the graph, you can ensure that the parabola is symmetric about this line.

When graphing, always reflect points across this line to maintain symmetry.

domain and range of a function

Understanding the domain and range of a function helps in identifying the input and output values a function can take.

***Domain:*** The domain is the set of all possible x-values (inputs) of the function. For any quadratic function in the form of a parabola, the domain is all real numbers because a parabola extends infinitely left and right. Thus, the domain for \( f(x) = -2(x+3)^2 + 4 \) is (-∞, ∞).

***Range:*** The range is the set of all possible y-values (outputs) of the function. Since our parabola opens downward (negative coefficient a = -2), the highest point is the vertex. The y-coordinate of the vertex is 4, which is the maximum value. Thus, the range is all y-values less than or equal to 4.

So, the range for the given function is \( (-∞, 4] \).

Knowing the domain and range helps you understand the extent of the graph both horizontally and vertically.