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Chapter 11: Problem 66

Graph each function. Be sure to label all the intercepts. $$f(x)=\sqrt{9-9 x^{2}}$$

Short Answer

Expert verified
Domain: \([-1, 1]\); Intercepts: \((0, 3)\), \((1, 0)\), \((-1, 0)\).

Step by step solution

01

Understanding the Function

The given function is \(f(x) = \sqrt{9 - 9x^2}\). This is a square root function involving a quadratic expression inside. Recognize that the expression inside the square root must be non-negative for the function to be defined.
02

Determine the Domain

To find the domain, set the expression inside the square root to be greater than or equal to zero: \[9 - 9x^2 \geq 0\] This simplifies to: \[1 - x^2 \geq 0\]This inequality holds when \(-1 \leq x \leq 1\). Therefore, the domain of the function is \([-1, 1]\).
03

Find the Intercepts

To find the y-intercept, substitute \(x = 0\) into the function:\[f(0) = \sqrt{9 - 9(0)^2} = \sqrt{9} = 3\]So the y-intercept is \((0, 3)\).For the x-intercepts, set the function equal to zero and solve for x: \[\sqrt{9 - 9x^2} = 0\]Square both sides: \[9 - 9x^2 = 0\]This simplifies to: \[9x^2 = 9\]\[x^2 = 1\]\[x = \pm 1\]So the x-intercepts are \((1, 0)\) and \((-1, 0)\).
04

Sketch the Graph

Plot the intercepts \((0, 3)\), \((1, 0)\), and \((-1, 0)\) on a coordinate plane. The function forms a half-ellipse (since it's derived from a quadratic expression inside a square root) that opens upwards and is bounded between \(-1\) and \(1\) on the x-axis. Draw a smooth curve connecting these points to illustrate the half-ellipse.
05

Label the Intercepts

Ensure the intercepts \((0, 3)\), \((1, 0)\), and \((-1, 0)\) are marked clearly on the graph. These points indicate where the function meets the x-axis and y-axis.

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Key Concepts

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

Quadratic Expressions
In the given exercise, we deal with the function \(f(x) = \sqrt{9 - 9x^2}\). The expression inside the square root, \(9 - 9x^2\), is a quadratic expression. Quadratic expressions are polynomials of degree 2, typically written in the form of \(ax^2 + bx + c\). In this case, the quadratic expression is simplified to \(9(1-x^2)\). The graph of \(9 - 9x^2\) is a downward-opening parabola. However, because the quadratic expression is inside a square root, we only consider the non-negative values, leading to the shape of a half-ellipse when graphed.
Domain of a Function
The domain of a function refers to all possible input values (x-values) that make the function defined and real. For the function \(f(x) = \sqrt{9 - 9x^2}\), the expression inside the square root, \((9 - 9x^2)\), must be non-negative (greater than or equal to zero) because the square root of a negative number is not a real number. To find this range of x-values, solve the inequality: \[9 - 9x^2 \geq 0\]
This simplifies to: \[1 - x^2 \geq 0\]
Next, recognize that \((1 - x^2)\) is non-negative when \(-1 \leq x \leq 1\). Thus, the function is defined for all x-values in the interval \([-1, 1]\), making this the domain of the function.
Intercepts
Intercepts are points where the graph of a function crosses the axes. For the function \(f(x) = \sqrt{9 - 9x^2}\), there are both y-intercepts and x-intercepts.
  • To find the y-intercept, set x to 0 and solve for f(x):\[f(0) = \sqrt{9} = 3\]
    The y-intercept is \((0, 3)\).

  • To find the x-intercepts, set the function equal to 0 and solve for x:\[\sqrt{9 - 9x^2} = 0\]
    Square both sides: \[9 - 9x^2 = 0\]
    Simplify to get: \[9x^2 = 9\]
    Which further simplifies to: \[x^2 = 1\]
    Thus, x can be \(\u00B1 1\), leading to x-intercepts at \((1, 0)\) and \((-1, 0)\).
Once you have these intercepts, you can plot them on the graph to help visualize the function's shape and behavior.

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