Problem 25 - 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. \(f(x)=4-(x-1)^{2}\)

Step-by-Step Solution

Answer

The vertex is (1, 4), the axis of symmetry is \(x=1\), the x-intercepts are \(x=-1\) and \(x=3\), and the y-intercept is \(y=3\). The domain is \(x \epsilon (-\infty, +\infty)\) and the range is \(y \epsilon (- \infty, 4] \).

Step 1: Identify the Vertex and Sketch

In the given quadratic function \( f(x) = 4 - (x - 1)^2 \), the vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where (h, k) is the vertex of the parabola. Here the vertex is (1, 4). The graph opens downward as 'a' is negative.

Step 2: Identify the Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that divides the parabola into two equal halves. It is given by the equation \( x = h \), where h is the x-coordinate of the vertex. For this parabola, the axis of symmetry is \( x = 1 \).

Step 3: Identify the Intercepts

To find the x-intercepts, set \( f(x) \) to zero and solve for x. Setting \( f(x) = 4 - (x - 1)^2 \) to zero gives the equation \( 4 - (x - 1)^2 = 0 \). Solving this equation yields \( x = 1 - \sqrt{4} \) and \( x = 1 + \sqrt{4} \), which simplify to \( x = -1 \) and \( x = 3 \). These are the x-intercepts. The y-intercept is found by plugging x=0 into the equation, yielding \( f(0) = 4 - (0-1)^2 = 3 \) which is the y-intercept.

Step 4: Determine the Domain and Range

The domain of a function is the set of all possible x-values. Because this is a quadratic function, and a parabola extends infinitely in the horizontal direction, the domain is \( x \epsilon (-\infty, +\infty) \). The range of a function is the set of all possible y-values. Because this parabola opens downward and its maximum point (the vertex) is at y=4, the range is \( y \epsilon (- \infty, 4] \).

Key Concepts

Vertex Form

The vertex form of a quadratic function is a special way of writing the equation that makes it easy to identify the key features of the parabola. The standard format for this is \( f(x) = a(x-h)^2 + k \), where \((h, k)\) represents the vertex of the parabola. In our function \( f(x) = 4 - (x-1)^2 \), we note that it's written as \( f(x) = -(x-1)^2 + 4 \).
This tells us right away that the vertex is at the point \((1, 4)\).

• The \( h \) value is 1, indicating the x-coordinate of the vertex.
• The \( k \) value is 4, which is the y-coordinate.

Since the coefficient \( a \) is negative (-1 in this case), the parabola opens downward. This means that the vertex provides the highest point on the graph, or the maximum value. Knowing the vertex helps in easily sketching the graph and understanding the behavior of the function.

Axis of Symmetry

The axis of symmetry is a key feature that helps balance a quadratic graph. This is a vertical line that runs through the vertex and divides the parabola into two mirror-image halves. For any quadratic function written in vertex form \( f(x) = a(x-h)^2 + k \), the axis of symmetry can be found with the equation \( x = h \).

In our example, since the vertex is \((1, 4)\), the axis of symmetry is simply \( x = 1 \). This vertical line at \( x = 1 \) acts as a mirror, meaning each point on one side of the parabola has a corresponding point directly across it on the other side. When sketching the parabola, this axis provides a crucial reference to ensure symmetry and accuracy.

Domain and Range

Understanding the domain and range of a quadratic function is crucial for grasping what the function fully encompasses. The domain of a function is all the possible x-values that can be input into the function. Since quadratic functions are defined for all real numbers, the domain for \( f(x) = 4 - (x - 1)^2 \) is \( x \in (-\infty, +\infty) \).

The range of a quadratic function is the set of y-values that the function can output. As the parabola here opens down, and its highest point is at the vertex \((1, 4)\), the range is capped at 4 and extends downwards. Thus, the range is \( y \in (-\infty, 4] \).

• The vertex provides the maximum y-value.
• The downward opening ensures that it drops to infinity on the y-axis.

Understanding the domain and range helps clarify the scope of the function's output and the behavior of the quadratic graph.

Intercepts

Intercepts are the points where the graph of a function crosses the x-axis and y-axis, providing essential information about the function's graph. In the quadratic function \( f(x) = 4 - (x-1)^2 \), we find the intercepts by setting the function to zero and substituting inputs accordingly.

To find the x-intercepts, set \( f(x) = 0 \).
Solving the equation \( 4 - (x-1)^2 = 0 \) gives the solutions \( x = -1 \) and \( x = 3 \), which are the points where the graph crosses the x-axis.
The y-intercept is found by substituting \( x = 0 \) into the function, yielding \( f(0) = 3 \). Hence, the graph crosses the y-axis at the point \((0, 3)\).

Knowing these intercepts allows us to plot critical points on the graph and is essential for accurately sketching the overall shape of the parabola.