Problem 19 - 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. \(f(x)=(x-1)^{2}+2\)

Step-by-Step Solution

Answer

The vertex is at (1,2), there is no x-intercept, and the y-intercept is at (0,3). The axis of symmetry is at \(x=1\). The domain of the function is all real numbers (-\(\infty\), \(\infty\)), and the range is [2, \(\infty\)).

Step 1: Identify the Vertex

The vertex form of a quadratic is \(f(x)=a(x-h)^{2}+k\), where (h,k) is the vertex. Comparing this with our given function \(f(x)=(x-1)^{2}+2\), we can say that the vertex of this quadratic function is (1,2).

Step 2: Find the intercepts

The x-intercepts are found by setting \(f(x)=0\). Hence, we solve the equation \((x-1)^{2}+2=0\). However, trying to solve this equation, we get a negative number under the square root which implies that there are no real roots. Thus, the function does not cross the x-axis, so there are no x-intercepts. Based on the vertex form, the y-intercept is gotten when \(x=0\). Substituting \(x=0\) in the function f(x), we get \(f(0)=(0-1)^{2}+2 = 3\). Hence the y-intercept is at (0,3).

Step 3: Plot the Vertex and Intercepts on a Graph

With the vertex at (1,2) and the y-intercept at (0,3), plot these points on a graph. Since there are no x-intercepts, the parabola doesn't cross the x-axis.

Step 4: Determine the Axis of Symmetry

The axis of symmetry of a parabola in vertex form \(f(x)=a(x-h)^{2}+k\) is the line \(x = h\). For our function \(f(x)=(x-1)^{2}+2\), the axis of symmetry is the line \(x=1\).

Step 5: Determine the Domain and Range

The domain of a quadratic function is always all real numbers which can be denoted as (-\(\infty\), \(\infty\)). The range of a quadratic function in the given form is \(y \geq k\) if the coefficient of \((x-h)^{2}\) is positive. Here the coefficient of \((x-1)^{2}\) is 1 (which is positive), so the range will be all y such that \(y \geq k\). Therefore, the range is [2, \(\infty\)) since the vertex's y-coordinate is 2.

Key Concepts

Vertex of a Quadratic Function

The vertex is the point where the parabola changes direction, either from increasing to decreasing or vice versa. In the vertex form of a quadratic function, which is given by \(f(x) = a(x-h)^2 + k\), the vertex is represented by the coordinates \((h, k)\). This makes it easy to identify the vertex just by looking at the equation itself. In our function \(f(x) = (x-1)^2 + 2\), comparing it with the vertex form reveals that the vertex is at the point \((1, 2)\).

• The vertex indicates the minimum or maximum point of the parabola.
• It provides a central point to plot the graph accurately.

Knowing the vertex helps in sketching the parabola correctly, as it shows the direction in which it opens and the highest or lowest point on the graph.

Intercepts of the Parabola

Intercepts are the points where the graph crosses the axes. For a quadratic function, these include the x-intercepts and y-intercepts.To find the x-intercepts, you set \(f(x) = 0\) and solve for \(x\). In our situation, solving \((x-1)^2 + 2 = 0\) results in no real solutions, indicating no x-intercepts for this function.The y-intercept, however, is found by setting \(x = 0\). Plugging in \(x = 0\), we get \(f(0) = (0-1)^2 + 2 = 3\). Thus, the y-intercept is at \((0, 3)\).

• X-intercepts may not always exist for quadratic functions, especially when no real roots are found.
• Y-intercepts occur where the function crosses the y-axis.

These points are critical as they provide guidance on the behavior and form of the graph.

Domain and Range of a Quadratic Function

The domain of a quadratic function is a set containing all possible \(x\)-values. For any quadratic, this is typically all real numbers, denoted as \((-\infty, \infty)\).The range, on the other hand, depends on the direction in which the parabola opens. For our equation \(f(x) = (x-1)^2 + 2\), the parabola opens upwards as the leading coefficient of \((x-1)^2\) is positive. Thus, the range begins at the minimum y-value at the vertex, which is \(2\), to infinity. This is represented as \([2, \infty)\).

• The domain always includes all real numbers for quadratic functions.
• The range depends on whether the parabola opens upwards or downwards.

Understanding domain and range is essential for determining the extent and behavior of the function graph.

Axis of Symmetry in Quadratic Functions

The axis of symmetry is a vertical line that passes through the vertex of the parabola, effectively dividing it into two mirror-image halves. For a quadratic function in vertex form \(f(x) = a(x-h)^2 + k\), the axis of symmetry is given by the line \(x = h\). In our function, this would be the line \(x = 1\).

• It helps in drawing the parabola by providing a central line of reference.
• The axis of symmetry ensures that any point on one side of the parabola has a corresponding point on the other side.

This concept is crucial as it not only aids in sketching but also helps understand the symmetry inherent in parabolic graphs.