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Magazine Chapter 1: Problem 23
Sketch the graph of \(f\) by hand. $$f(x)=x^{2}$$
Short Answer
Expert verified
The graph of \(f(x) = x^2\) is a parabola opening upwards with vertex at (0,0).
Step by step solution
01
Identify the Type of Function
The function given is \(f(x) = x^2\), which is a basic quadratic function. This means it will be a parabola.
02
Identify Key Features of the Parabola
The vertex of the parabola \(f(x) = x^2\) is at the origin, \((0,0)\). Since the coefficient of \(x^2\) is positive, the parabola opens upwards.
03
Determine the Axis of Symmetry
The axis of symmetry for the function \(f(x) = x^2\) is the y-axis, which corresponds to the line \(x = 0\).
04
Find the Intercepts
The y-intercept occurs at \(f(0) = 0^2 = 0\), hence the graph passes through the point \((0,0)\). As the function is symmetric, this point is also the only x-intercept.
05
Plot Additional Points
Plot additional points to help sketch the graph, such as \((-1,1)\), \((1,1)\), \((-2,4)\), \((2,4)\), and \((-3,9)\), \((3,9)\). Compute these by substituting \(x\) values into \(f(x).\)
06
Draw the Parabola
Using the plotted points, sketch a U-shaped curve opening upwards, making sure it is symmetric about the y-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graph Sketching
Sketching the graph of a quadratic function by hand can be both insightful and fun. The goal is to visualize how the function behaves across different values of \(x\). A quadratic function like \(f(x) = x^2\) always forms a specific shape called a parabola. To create a sketch, you want to plot key points by choosing a range of \(x\) values. Doing so helps establish the general shape.
- First, identify the function type. In this case, \(f(x) = x^2\) is a standard quadratic function.
- Next, determine symmetry. This can simplify plotting additional points once you know one side of the graph.
Parabola
A parabola is a unique curve represented by quadratic equations like \(f(x) = ax^2 + bx + c\). For the basic function \(f(x) = x^2\), the parabola has a few distinct features: it is symmetrical, U-shaped, and centers around a point known as the vertex. The parabola's arms can be upwards or downwards depending on the sign of the \(a\) coefficient in front of \(x^2\). Here, because \(a\) is positive in \(f(x) = x^2\), the parabola opens upwards.
Understanding the nature of a parabola helps to predict how the graph will look without plotting every single point.
Understanding the nature of a parabola helps to predict how the graph will look without plotting every single point.
- It can be stretched or compressed vertically by changing the coefficient \(a\).
- It shifts along the x or y axis with non-zero \(b\) and \(c\) coefficients.
Axis of Symmetry
For every parabola, there is an axis of symmetry. This is an imaginary vertical line that splits the parabola into two mirror-image halves. For the function \(f(x) = x^2\), the axis of symmetry is the line \(x = 0\). It coincides with the y-axis. Knowing the axis of symmetry is crucial because it:
- Helps quickly find that the graph is balanced on both sides.
- Assists in plotting points efficiently since any point \((x, y)\) has a corresponding \((-x, y)\) point.
Intercepts
Intercepts are where the graph crosses the coordinate axes and are vital for sketching graphs. The y-intercept for the function \(f(x) = x^2\) occurs at the origin \((0,0)\), where \(x = 0\). You can find this easily by substituting \(x = 0\) into the function: \(f(0) = 0^2 = 0\). This is the only intercept for the basic function, as the parabola also opens upwards around this point.
Since it is symmetric, it doesn't cross the x-axis anywhere else. Intercepts give you precise anchor points for sketching and are the simplest points to identify on a graph.
Since it is symmetric, it doesn't cross the x-axis anywhere else. Intercepts give you precise anchor points for sketching and are the simplest points to identify on a graph.
Vertex of a Parabola
The vertex is the peak or the lowest point of the parabola, depending on whether it opens downward or upward. For \(f(x) = x^2\), the vertex is at \((0,0)\).
- When a parabola opens upwards, the vertex is the lowest point.
- Conversely, for downward opening parabolas, it's the highest point.
