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Chapter 2: Problem 13

Sketching Graphs of Quadratic Functions In }} \\ {\text { Exercises } 13-16, \text { sketch the graph of each quadratic }} \\ {\text { function and compare it with the graph of } y=x^{2} .}\end{array} $$ \begin{array}{ll}{\text { (a) } f(x)=\frac{1}{2} x^{2}} & {\text { (b) } g(x)=-\frac{1}{8} x^{2}} \\ {\text { (c) } h(x)=\frac{3}{2} x^{2}} & {\text { (d) } k(x)=-3 x^{2}}\end{array} $$

Short Answer

Expert verified
The shape of these function's graphs is similar to \(y=x^{2}\), either mirrored on the x-axis, or vertically stretched or compressed due to the coefficient of \(x^{2}\). The graphs can be sketched respectively.

Step by step solution

01

Sketching the graph of \(f(x)=\frac{1}{2} x^{2}\)

Here, the coefficient of \(x^{2}\) is \(\frac{1}{2}\), which is less than 1. This means the graph of \(y=x^{2}\) will compress vertically to get the graph of \(f(x)=\frac{1}{2} x^{2}\).
02

Sketching the graph of \(g(x)=-\frac{1}{8} x^{2}\)

In this case, the negative sign before the coefficient of \(x^{2}\) indicates a reflection over the x-axis. The coefficient \(\frac{1}{8}\) is less than 1, so the graph of \(y=x^{2}\) will also compress vertically to get the graph of \(g(x)=-\frac{1}{8} x^{2}\).
03

Sketching the graph of \(h(x)=\frac{3}{2} x^{2}\)

Here, the coefficient of \(x^{2}\) is \(\frac{3}{2}\), which is greater than 1. This means the graph of \(y=x^{2}\) will stretch vertically to get the graph of \(h(x)=\frac{3}{2} x^{2}\).
04

Sketching the graph of \(k(x)=-3 x^{2}\)

In this case, the negative sign before the coefficient of \(x^{2}\) indicates a reflection over the x-axis. The coefficient 3 is greater than 1, so the graph of \(y=x^{2}\) will also stretch vertically to get the graph of \(k(x)=-3 x^{2}\).

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Key Concepts

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

Vertical Stretch and Compression
Understanding how a quadratic function graph stretches or compresses vertically is key to mastering quadratic functions. Let's explore what makes a graph stretch or compress. The vertical stretch or compression is dictated by the absolute value of the coefficient in front of the term with \(x^2\).

  • If the absolute value of the coefficient is greater than 1, the graph experiences a vertical stretch.
  • If the absolute value of the coefficient is less than 1, the graph undergoes a vertical compression.
To put it simply, stretching makes the graph "narrower," while compressing makes it "wider." This change is because the function's values grow faster or slower compared to those of \(y = x^2\).
For example:
  • In \(f(x)=\frac{1}{2} x^2\), since \(\frac{1}{2}\) is less than 1, the graph compresses vertically, making it wider than \(y = x^2\).
  • In \(h(x)=\frac{3}{2} x^2\), since \(\frac{3}{2}\) is more than 1, the function stretches vertically, resulting in a narrower graph.
Reflection over the x-axis
A quadratic function can also be reflected over the x-axis, a concept that flips the graph upside down. But how do we know if a graph reflects over the x-axis? This happens when the coefficient of the \(x^2\) term is negative.

Let's consider why this reflection occurs:
  • When the coefficient is positive, the graph opens upwards, creating a U-shape.
  • When the coefficient is negative, the graph opens downwards, creating an upside-down U-shape.
This change doesn't affect the degree of stretch or compression鈥攋ust the orientation.
For instance:
  • In \(g(x)=-\frac{1}{8} x^2\), the negative sign indicates a reflection over the x-axis, which flips the graph downwards compared to \(y = x^2\).
  • Similarly, \(k(x)=-3 x^2\) reflects over the x-axis, making its graph open downwards as well.
Coefficient of x虏
The coefficient of \(x^2\) carries important information about the graph of a quadratic function. It determines both the direction and the vertical nature of the graph. This coefficient is crucial for understanding quadratic transformations.

Here's what it tells us:
  • The sign of the coefficient reveals whether the graph opens upwards or downwards. Positive means upwards, negative means downwards.
  • The magnitude of the coefficient informs us about the vertical stretch or compression.
So, examining the coefficient allows us to capture the essence of the parabola's shape and orientation.
Consider these examples:
  • In \(f(x)=\frac{1}{2} x^2\), the positive \(\frac{1}{2}\) means the graph opens upwards and compresses vertically.
  • In \(k(x)=-3 x^2\), the negative 3 means the graph opens downwards and stretches vertically, providing a dramatic inverted U-shape.
Understanding these elements is key to sketching and interpreting graphs of quadratic functions.

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