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Chapter 0: Problem 11

Determine the intercepts of the graphs of the following equations. $$f(x)=9 x+3$$

Short Answer

Expert verified
(0, 3) and $$ \left( - \frac{1}{3}, 0 \right) $$.

Step by step solution

01

- Find the y-intercept

To find the y-intercept of the function, set the value of x to zero and solve for f(x). Substitute x with 0 in the equation: equation:\[ f(x) = 9x + 3 \] \[ f(0) = 9(0) + 3 \] \[ f(0) = 3 \] The y-intercept is the point (0, 3).
02

- Find the x-intercept

To find the x-intercept of the function, set f(x) to zero and solve for x. equation:$$ f(x) = 9x + 3 $$ Set $$ f(x) = 0 $$: $$ 0 = 9x + 3 $$ Solve for x: $$ 9x = -3 $$ $$ x = -\frac{3}{9} $$ $$ x = -\frac{1}{3} $$ The x-intercept is the point $$ \left( - \frac{1}{3}, 0 \right) $$.

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Key Concepts

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

linear equations
Linear equations are fundamental in algebra. They represent relationships where the outcome is a straight line when graphed. In the form of a linear equation like \[ y = mx + b \], 'm' is the slope, and 'b' is the y-intercept. The equation provided \[ f(x) = 9x + 3 \] fits this form. Here, 9 is the slope, indicating how steep the graph is, and 3 is the point where the line intersects the y-axis. Linear equations are straightforward but powerful tools in algebra, allowing us to model and understand various relationships.
y-intercept
The y-intercept of a graph is the point where it crosses the y-axis. To find the y-intercept in a linear equation, set x to 0 and solve for y. For the equation \[ f(x) = 9x + 3 \], substitute x with 0:
\[ f(0) = 9(0) + 3 \]
\[ f(0) = 3 \]
Thus, the y-intercept is the point (0, 3). This is where the line touches the y-axis, giving us a clear starting point for graphing the function. Remember, it鈥檚 always a good first step to identify the y-intercept when dealing with linear equations.
x-intercept
The x-intercept is the point where the graph crosses the x-axis. To find this, set f(x) to 0 and solve for x. For the equation \[ f(x) = 9x + 3 \], set f(x) to 0:
\[ 0 = 9x + 3 \]
Now, solve for x:
\[ 9x = -3 \]
\[ x = -\frac{3}{9} \]
\[ x = -\frac{1}{3} \]
The x-intercept is the point \( -\frac{1}{3}, 0 \). This helps us understand where the line meets the x-axis, providing another key point for graphing.
graphing functions
Graphing functions can be fun and insightful. To graph a linear equation, start by plotting both intercepts. For instance, with \[ f(x) = 9x + 3 \], plot the y-intercept at (0, 3) and the x-intercept at \( -\frac{1}{3}, 0 \). Draw a straight line through these points.
  • The intercepts give you the basic framework.
  • The slope (in this case, 9) tells you how steep the line is.
It's that simple! Understanding intercepts and slopes can turn any linear equation into an easy-to-draw graph, making abstract algebraic concepts visual and clear.

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Most popular questions from this chapter

Use your graphing calculator to find the value of the given function at the indicated values of \(x .\) $$f(x)=x^{4}+2 x^{3}+x-5 ; \quad x=-\frac{1}{2}, x=3$$

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The expressions may be factored as shown. Find the missing factors. $$x^{-1 / 4}+6 x^{1 / 4}=x^{-1 / 4}(\quad)$$

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\frac{\left(-27 x^{5}\right)^{2 / 3}}{\sqrt[3]{x}}$$

Let \(f(x)=\sqrt[3]{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Calculate the following functions. Take \(x > 0\). $$[f(x) g(x)]^{3}$$
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