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Chapter 11: Problem 42

For each of the following, graph the function, label the vertex, and draw the axis of symmetry. $$ f(x)=2(x+7)^{2} $$

Short Answer

Expert verified
Graph has a vertex at (-7,0) and axis of symmetry at x = -7.

Step by step solution

01

- Identify the Vertex

The function is in vertex form, which is given by: \text{\(f(x)=a(x-h)^{2}+k\)} Comparing it to \(f(x)=2(x+7)^{2}\), we find \(h=-7\) and \(k=0\). Therefore, the vertex of the function is at \((-7,0)\).
02

- Graph the Function

Plot the vertex \((-7,0)\) on the coordinate plane. The function \(f(x)=2(x+7)^{2}\) is a parabola that opens upwards (because the coefficient of \((x+7)^{2}\) is positive). For more accuracy, plot additional points by selecting values of \(x\) around -7 and calculating corresponding \(f(x)\) values (e.g., \(f(-8)\), \(f(-6)\)). Draw a smooth curve passing through these points and the vertex.
03

- Draw the Axis of Symmetry

The axis of symmetry for a parabola in vertex form \(f(x)=a(x-h)^{2}+k\) is the vertical line that passes through the vertex. For the function \(f(x)=2(x+7)^{2}\), the axis of symmetry is the line \(x=-7\). Draw a dashed vertical line at \(x=-7\) on the graph to represent the axis of symmetry.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vertex form
The vertex form of a quadratic function is a helpful format for graphing. It shows the vertex and opens the formula quickly. The general form for vertex form is given by:
\text{\(f(x)=a(x-h)^{2}+k\)}
Here, \text{\(h\)} and \text{\(k\)} are the coordinates of the vertex. The \text{\(a\)} represents the stretch factor. In our function \text{\(f(x)=2(x+7)^{2}\)}, comparing it with the general form, we see \text{\(h=-7\)} and \text{\(k=0\)}. Therefore, the vertex of this function is \text{\((-7,0)\)}. Knowing the vertex helps us position the parabola on the coordinate plane.
parabola
A parabola is a U-shaped curve, and it鈥檚 the graph resulting from a quadratic function. The simplest parabola is \text{\(y=x^{2}\)}, which opens upwards. When a quadratic function is given in vertex form, \text{\(f(x)=a(x-h)^{2}+k\)}, it describes a parabola with vertex at \text{\((h,k)\)}.
  • If \text{\(a\)} is positive, the parabola opens upwards.
  • If \text{\(a\)} is negative, the parabola opens downwards.
For our function, \text{\(f(x)=2(x+7)^{2}\)}, the coefficient 2 is positive, so this parabola opens upwards. We can visualize it by plotting additional points and sketching the curve passing through them and the vertex.
axis of symmetry
The axis of symmetry is an imaginary line that divides the parabola into two mirror-image halves. For a quadratic function in vertex form \text{\(f(x)=a(x-h)^{2}+k\)}, the axis of symmetry is always the vertical line \text{\(x=h\)}.
In our example with \text{\(f(x)=2(x+7)^{2}\)}, since \text{\(h=-7\)}, the axis of symmetry is the vertical line \text{\(x=-7\)}. Drawing this line on the graph helps to visualize and confirm the symmetry of the parabola.
vertex
The vertex of a parabola is its highest or lowest point, depending on its orientation. A quadratic function in vertex form easily reveals the vertex at \text{\((h,k)\)}. In our case, the function \text{\(f(x)=2(x+7)^{2}\)} has the vertex at \text{\((-7,0)\)}.
  • The vertex is always a turning point for the graph.
  • It represents the maximum or minimum value of the function, depending on whether the parabola opens downwards or upwards.
Plotting the vertex is crucial for graphing the overall shape and position of the parabola accurately.
coordinate plane
The coordinate plane is a grid that helps to graph equations and functions. It consists of two axes: the horizontal (x-axis) and the vertical (y-axis). The point where they intersect is the origin \text{\((0,0)\)}. When graphing a quadratic function:
  • Plot the vertex first on the coordinate plane.
  • Use the axis of symmetry to guide additional point placement.
  • Mark key points, plot them, and sketch the curve.
For \text{\(f(x)=2(x+7)^{2}\)}, start by plotting \text{\((-7,0)\)}. Then, calculate values of \text{\(f(x)\)} for \text{\(x\)} near -7, such as -6 and -8, and plot these points to help sketch the upward-opening parabola accurately.

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