Problem 23 - College Algebra

Question

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)-2(x+2)^{2}-1$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = -2(x+2)^2 -1\) is a downward-opening parabola with the vertex at \((-2, -1)\), x-intercepts at \(-3\) and \(-1\), y-intercept at \(-3\). Its axis of symmetry is the vertical line \(x = -2\). The domain of the function is all real numbers (\(-\infty, +\infty\)) while the range is \(-\infty, -1]\).

Step 1. Identify and Plot the Vertex

The vertex (\(h\), \(k\)) of the function \(f(x) = -2(x+2)^2 -1\) is (-2, -1). Plot this point on the graph.

Step 2. Find and Plot the Intercepts

To find the x-intercepts, set \(y = f(x) = 0\) and solve for \(x\) (note that these are the points where the graph meets the x-axis). \n\nAnd the y-intercept is found by finding \(f(0)\). \n\nSolving \(f(x) = 0\), we get \(x = -1\) and \(-3\). \n\nEvaluating \(f(0)\), we get \(y = -3\).\n\nPlot the x-intercepts (\(-1, 0\) and \(-3, 0\)) and the y-intercept \((0, -3)\) on the graph.

Step 3: Draw the Parabola

Having the vertex and intercepts, draw a smooth curve to create the graph of the function. Since \(a = -2 < 0\), the parabola opens downwards.

Step 3: Define Axis of Symmetry

The axis of symmetry of a parabola is the vertical line through the vertex. In this case, it is \(x = -2\).

Step 4: Determine Domain and Range

The domain of a function is the set of all possible x-values. For any quadratic function, this is all real numbers, so the domain is \(-\infty, +\infty\).\n\nThe range of a function is the set of all possible y-values. Since our parabola opens downwards and its highest point is at the vertex, the max value of y is -1 and it goes downwards to negative infinity. So the range is \(-\infty, -1]\)

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial for sketching its graph accurately. The vertex is the highest or lowest point on a parabola, depending on whether it opens upward or downward. For the function f(x) = -2(x+2)^2 -1, the vertex is identified by reworking the equation into its completed square form, which directly reveals the vertex as (h, k). In this function, the vertex is at (-2, -1). Since the coefficient of x2 is negative (-2), the parabola opens downwards, making this vertex the parabola's maximum point.

Importance of the Vertex

• It helps in determining the parabola's orientation (upwards or downwards).
• It is essential for finding the axis of symmetry.
• The vertex aids in identifying the parabola's maximum or minimum value, which is crucial for understanding the parabola's range.

Intercepts of Quadratic Function

The intercepts of a quadratic function are the points where the graph crosses the x-axis and y-axis, known as x-intercepts and y-intercepts respectively. These points are incredibly useful when sketching the graph of a quadratic function.

To find the x-intercepts, we search for values of x that make f(x) equal to zero. For the function f(x) = -2(x+2)^2 -1, solving f(x) = 0 gives us x-intercepts at x = -1 and x = -3. The y-intercept occurs where x=0; plugging 0 into the function yields the y-intercept at y = -3.

Significance of Intercepts

• X-intercepts can indicate the number of real roots the quadratic equation has.
• Y-intercept provides a starting point for graphing the parabola.
• Intercepts are key reference points in constructing the shape of the graph.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that divides it into two mirror-images, ensuring that each side is a reflection of the other. This axis passes through the vertex of the parabola. For our function f(x) = -2(x+2)^2 -1, the axis of symmetry is the line x = -2. It provides a valuable reference for graphing the parabola since every point on one side of the axis can be reflected across it to find a corresponding point on the other side.

Key Points about Axis of Symmetry

• The formula to find the axis of symmetry for a quadratic function in standard form (ax^2 + bx + c) is x = -b/(2a).
• It helps in simplifying the graphing process by providing a 'mirror line'.
• Knowing the axis of symmetry is essential in determining the direction of the parabola's opening.

Domain and Range of a Function

The domain of a function consists of all the possible input values (x-values), whereas the range includes all possible output values (y-values). A quadratic function, being a polynomial, has a domain of all real numbers, written as (-

For the given function f(x) = -2(x+2)^2 -1, the domain is the set of all real numbers ((-

The range, however, is dependent on the direction the parabola opens and its vertex. Since our parabola opens downward with a vertex at (-2, -1), the maximum y-value is -1, and it extends to negative infinity, giving us a range of (-

Understanding Domain and Range

• The domain of any quadratic function is (-
  
• For a downward-opening parabola, the range is typically (-
  
• The vertex's y-coordinate sets the boundary for the range in a quadratic function.