Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the quadratic function. Identify the vertex and intercepts. $$f(x)=(x-6)^{2}+3$$

Step-by-Step Solution

Answer

The vertex of the quadratic function \(f(x) = (x - 6)^2 + 3\) is at (6,3). The y-intercept is at (0,39) and there are no x-intercepts. The graph of the function opens upward and lies above the x-axis.

Step 1: Identify the Vertex

The vertex form of a quadratic function is \(f(x) = a(x - h)^2 + k\), where (h,k) is the vertex. In the given function \(f(x) = (x - 6)^2 + 3\), you can observe that h = 6 and k = 3. Therefore, the vertex of the parabola is (6,3).

Step 2: Find the Y-intercept

To find the y-intercept, set x = 0 in the quadratic function. The y-intercept is the point where the graph intersects the y-axis. In this function, setting x=0 gives, \(f(0) = (0 - 6)^2 + 3 = 39\), so the y-intercept is (0, 39).

Step 3: Find the X-intercepts

To find the x-intercepts, set \(f(x) = 0\) and solve for x. The x-intercepts are the points where the graph intersects the x-axis. Setting \(f(x) = 0\) gives us \((x - 6)^2 + 3 = 0\). Solving this, we realize there are no real roots as the minimum value of \((x - 6)^2\) is 0 and the equation \((x - 6)^2 = -3\) has no real solutions. Thus, there are no x-intercepts.

Step 4: Sketch the Graph

Now with the vertex and intercepts, you can sketch the graph. Plot the vertex at (6,3). Then plot the y-intercept at (0,39). Since there are no x-intercepts, the parabola does not intersect the x-axis. The graph opens upwards as the coefficient of \((x - 6)^2\) is positive, and because there is no x-intercept, the graph lies above the x-axis.

Key Concepts

Vertex of a Parabola

In a quadratic function, the vertex is often considered the most crucial point on the graph. It either represents the highest or lowest point, depending on whether the parabola opens upwards or downwards. For the quadratic function in vertex form, \(f(x) = a(x-h)^2 + k\), the vertex is located at the coordinate \((h, k)\).
For the given function \(f(x) = (x-6)^2 + 3\):

• The vertex is \((6, 3)\).
• This point indicates the lowest point of the parabola since the parabola opens upwards.

The vertex form makes it easy to identify the vertex directly from the equation without any complex calculations.

Y-intercept

The y-intercept is the point where a graph crosses the y-axis. To find it, you set \(x = 0\) and solve for \(f(x)\). This gives the point \((0, f(0))\) on the graph.
For the given quadratic function, substitute \(x = 0\) into \(f(x) = (x-6)^2 + 3\) to find the y-intercept:

• Evaluate \((0-6)^2 + 3\), which simplifies to 39.
• So, the y-intercept is \((0, 39)\).

This point helps in sketching the graph, especially if the parabola doesn't intersect the x-axis.

X-intercepts

X-intercepts, also known as roots or zeroes, are where the graph crosses the x-axis. Finding them involves solving \(f(x) = 0\) for \(x\). These intercepts are solutions to the quadratic equation.
For the function \(f(x) = (x-6)^2 + 3\), solve:

• Set \((x-6)^2 + 3 = 0\).
• This simplifies to \((x-6)^2 = -3\), which has no real solutions.
• This means the parabola does not intersect the x-axis.

Lack of x-intercepts indicates that the entire graph of this function lies above the x-axis, illustrating that no real solutions exist.

Graph Sketching

Graph sketching of a quadratic function combines understanding the vertex, intercepts, and the direction the parabola opens. In our quadratic function, after identifying all this information, we can draw a rough graph.
Steps to sketch the graph:

• Start by plotting the vertex at \((6, 3)\).
• Next, include the y-intercept at \((0, 39)\).
• Since there are no x-intercepts, the parabola does not touch the x-axis at any point.
• The parabola opens upwards, as indicated by the positive coefficient of \((x-6)^2\).
• Finally, ensure that the entire graph remains above the x-axis.

By combining this information, you can sketch an accurate representation of the quadratic function’s path, even with only the vertex and y-intercept available.