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Chapter 4: Problem 95

Graph each equation by completing the table of values. $$\begin{aligned} &y=x^{2}-4\\\&\begin{array}{c|c}\hline x & y \\\\\hline-2 & \\\\\hline-1 & \\\\\hline 0 & \\\\\hline 1 & \\\\\hline 2 &\end{array}\end{aligned}$$

Short Answer

Expert verified
-2, 0; -1, -3; 0, -4; 1, -3; 2, 0

Step by step solution

01

- Understand the Equation

The given quadratic equation is \(y = x^2 - 4\). This equation tells us that for each value of \(x\), we can find the corresponding \(y\) by squaring \(x\) and then subtracting 4.
02

- Fill in the Table for \(x = -2\)

When \(x = -2\), substitute it into the equation: \(y = (-2)^2 - 4 = 4 - 4 = 0\). So, \y = 0\.
03

- Fill in the Table for \(x = -1\)

When \(x = -1\), substitute it into the equation: \(y = (-1)^2 - 4 = 1 - 4 = -3\). So, \y = -3\.
04

- Fill in the Table for \(x = 0\)

When \(x = 0\), substitute it into the equation: \(y = (0)^2 - 4 = 0 - 4 = -4\). So, \y = -4\.
05

- Fill in the Table for \(x = 1\)

When \(x = 1\), substitute it into the equation: \(y = (1)^2 - 4 = 1 - 4 = -3\). So, \y = -3\.
06

- Fill in the Table for \(x = 2\)

When \(x = 2\), substitute it into the equation: \(y = (2)^2 - 4 = 4 - 4 = 0\). So, \y = 0\.
07

- Complete the Table

Now we have calculated the \(y\) values for each \(x\). The completed table is: \ \ \ \ \[\begin{array}{c|c} \hline x & y \hline-2 & 0 \hline-1 & -3 \hline 0 & -4 \hline 1 & -3 \hline 2 & 0\end{array}\] \
08

- Plot the Points

To graph the equation, plot the points on a coordinate plane: \((-2, 0)\), \((-1, -3)\), \((0, -4)\), \((1, -3)\), and \((2, 0)\).
09

- Draw the Parabola

Connect the points with a smooth, curved line to complete the graph of the quadratic equation \(y = x^2 - 4\). This line is called a parabola.

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Key Concepts

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

quadratic function
A quadratic function is a type of polynomial function with the form: \( y = ax^2 + bx + c \).In this equation, \( a \), \( b \), and \( c \) are constants, with \( a \) not equal to zero. The quadratic function always gives a parabolic shape when graphed.For the specific exercise, we worked with the equation \( y = x^2 - 4 \). Here, the coefficients are:
  • \( a = 1 \)
  • \( b = 0 \)
  • \( c = -4 \)
This equation shows that as \( x \) changes, \( y \) is determined by squaring \( x \) and subtracting 4. The equation does not have a linear term (\( bx \)) since \( b = 0 \).
table of values
Filling out a table of values helps to understand the function better by providing specific outputs for given inputs. For the quadratic equation \( y = x^2 - 4 \), we chose different \( x \) values to find corresponding \( y \) values.By calculating, we found:
  • For \( x = -2 \), \( y = 0 \)
  • For \( x = -1 \), \( y = -3 \)
  • For \( x = 0 \), \( y = -4 \)
  • For \( x = 1 \), \( y = -3 \)
  • For \( x = 2 \), \( y = 0 \)
Using these values, we can then plot these points on a graph to visualize the relationship between \( x \) and \( y \). This method often clarifies how the function behaves.
parabola
A parabola is the U-shaped curve we get when we graph a quadratic function. The direction (opening upwards or downwards) of the parabola depends on the coefficient \( a \) in the quadratic function.
  • If \( a > 0 \), the parabola opens upwards.
  • If \( a < 0 \), it opens downwards.
For the exercise we did, since \( a = 1 \), the parabola for \( y = x^2 - 4 \) opens upwards. To graph it, we plot the points obtained from our table of values and then connect them smoothly. This helps to visually understand the shape and position of the parabola.

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Most popular questions from this chapter

Perform each indicated operation. \(\left(-2 b^{6}+3 b^{4}-b^{2}\right)+\left(b^{6}+2 b^{4}+2 b^{2}\right)\)

Without actually performing the operations, mentally determine the coefficient of the \(x^{2}\) -term in the simplified form of $$\left(-4 x^{2}+2 x-3\right)-\left(-2 x^{2}+x-1\right)+\left(-8 x^{2}+3 x-4\right)$$

Graph each equation by completing the table of values. $$ \begin{aligned} &y=2 x^{2}+2\\\ &\begin{array}{c|c} \hline x & y \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \end{array} \end{aligned} $$

If one "dog" year is estimated to be about seven "human" years, the monomial \(7 x\) gives the dog's age in human years for \(x\) dog years. Evaluate this monomial for \(x=9\). Use the result to fill in the blanks: If a dog is ______ in dog years, then it is _________ in human years.

Use scientific notation to calculate the result in each expression. Write answers in scientific notation. 74\. \(\frac{3.400,000,000(0.000075)}{0.00025}\)
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