Question

Graph the function. Label the vertex, axis of symmetry, and \(x\)-intercepts. \(g(x)=6(x-4)^2\)

Short answer

The vertex of the function is at (4, 0), the axis of symmetry is \(x = 4\), the function has one x-intercept at (4,0). The graph is a parabola opening upwards and symmetric about the line \(x = 4\).

Step-by-step solution

Find the Vertex and Axis of Symmetry

The function \(g(x)=6(x-4)^2\) is in the form of \(a(x-h)^2\), with a = 6 and h = 4. This means the vertex of the parabola is at (h, 0) = (4,0). The axis of symmetry is the vertical line through the vertex, so the equation of the axis of symmetry is \(x = h = 4\).

Find the x-intercepts

The \(x\)-intercepts of the function are the values of \(x\) such that \(g(x) = 0\). Setting \(g(x) = 0\) gives the equation \(6(x - 4)^2 = 0\). Solving this quadratic equation gives \(x = 4\) as the only solution, which means the graph of the function touches the x-axis at (4,0).

Graph the Function

First draw the axis of symmetry \(x = 4\). Then, plot the vertex at (4, 0) and the \(x\)-intercept also at (4,0). Since the coefficient of \((x - 4)^2\) is positive, the parabola opens upwards. The final graph is a parabola that touches the x-axis at the vertex, opens upwards and is symmetric about the line \(x = 4\).

Key concepts

Graphing Parabolas

Graphing a parabola involves understanding its shape and key features, such as the vertex, axis of symmetry, and intercepts. A parabola is a U-shaped curve that can open either upwards or downwards. Whether it opens up or down depends on the sign in front of the quadratic term. In the function \(g(x) = 6(x - 4)^2\), the coefficient of the squared term is positive (6), indicating that the parabola opens upwards.

To graph a parabola effectively:

• Identify and plot the vertex, which is the turning point of the parabola.
• Draw the axis of symmetry, a vertical line that splits the parabola in half.
• Find and plot the x-intercepts, which are the points where the parabola crosses the x-axis.

Combining these elements gives you a clear picture of the parabola's behavior on a graph.

Vertex of a Parabola

The vertex of a parabola is its peak or lowest point, depending on whether the parabola opens downwards or upwards, respectively. In the standard form \(g(x) = 6(x - 4)^2\), the vertex is easily identified: it is located at the point \((h, k)\), where \(h\) and \(k\) are derived from the equation.

For our function, \(h\) is 4 and \(k\) is 0, indicating that the vertex is at the point (4, 0). The vertex serves as the starting reference for graphing and helps us understand how the parabola is positioned relative to the x-axis.

Plotting the vertex on a graph is the first step in sketching the parabola. Since our parabola does not move up or down (signified by \(k\) being zero), the vertex lies directly on the x-axis at this point.

Axis of Symmetry

The axis of symmetry is a crucial line that divides the parabola into two identical halves. This symmetry line passes vertically through the vertex.

For the function \(g(x) = 6(x - 4)^2\), the axis of symmetry can be determined from the expression \(x = h\), where \(h = 4\). Therefore, the axis of symmetry is \(x = 4\). This means that for every point on one side of the axis, there is an identical point on the opposite side.

Marking the axis of symmetry on your graph helps you ensure that the parabolic shape is balanced, which is important for both accurate graphing and understanding the function's behavior. It's like a mirror line for the parabola.

X-Intercepts

X-intercepts are where the graph of a function crosses or touches the x-axis. They represent the values of \(x\) for which the function value \(g(x)\) is zero.

In \(g(x) = 6(x - 4)^2\), setting the function to zero, we solve the equation \(6(x - 4)^2 = 0\). Solving this gives us \(x - 4 = 0\), or \(x = 4\). There is only one x-intercept at (4, 0), which also happens to be the vertex.

This is a unique case where the parabola just touches the x-axis at the vertex, indicating it has no other real roots. Thus, the parabola does not cross the x-axis; it merely touches it.