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Chapter 5: Problem 1

The intercepts of the graph of \(9 x^{2}+4 y=36\) are ______.

Short Answer

Expert verified
The intercepts are (2, 0), (-2, 0), and (0, 9).

Step by step solution

01

Identify the intercepts

The intercepts of a graph are the points where the graph crosses the x-axis and the y-axis.
02

Find the x-intercepts

To find the x-intercepts, set y to 0 in the equation and solve for x. Given the equation: \[9 x^{2}+4 y = 36\], set y = 0 to get: \[9 x^{2} + 4 \times 0 = 36\] \[9 x^{2} = 36\] \[x^{2} = 4\] \[x = \text{卤2}\] The x-intercepts are at (2, 0) and (-2,0).
03

Find the y-intercepts

To find the y-intercepts, set x to 0 in the equation and solve for y. Given the equation: \[9 x^{2}+4 y = 36\], set x = 0 to get: \[9 \times 0^{2} + 4 y = 36\] \[4 y = 36\] \[y = 9\] The y-intercept is at (0, 9).
04

Combine the intercepts

Combine the solutions to report the intercepts. The x-intercepts are at (2, 0) and (-2,0), and the y-intercept is at (0, 9).

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Key Concepts

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

x-intercepts
The x-intercepts of an equation are the points where the graph crosses the x-axis. To find x-intercepts, we set the value of y to zero in the given equation. For the equation \[9x^2 + 4y = 36\], setting y to zero gives us \[9x^2 = 36\]. By solving this equation, we find that \[x = \text{卤2}\]. Therefore, the x-intercepts are at the points (2, 0) and (-2, 0). These points indicate where the graph meets the x-axis.
y-intercepts
The y-intercepts of a graph are the points where the graph crosses the y-axis. To find the y-intercept, we set the value of x to zero in the given equation. Using the equation \[9x^2 + 4y = 36\], setting x to zero gives us \[4y = 36\]. Solving this equation, we get \[y = 9\]. Thus, the y-intercept is at the point (0, 9). This is where the graph meets the y-axis.
solving quadratic equations
Solving quadratic equations like \[9x^2 + 4y = 36\] involves isolating the variable we are solving for. In the case of finding intercepts, we either set x or y to zero and then solve the resulting equation. A quadratic equation in the form \[ax^2 + by = c\] can be solved by isolating x or y and using algebraic manipulation to find the square root. This is done by:
  • Isolating the quadratic term (e.g., \[9x^2 = k\] or \[4y = k\])
  • Solving for the variable (e.g., \[x = \text{卤}\frac{\text{Root of k}}{a}\] or \[y = \frac{k}{b}\])
This process helps find the points where the graph intersects axes.
graphing
Graphing a quadratic equation like \[9x^2 + 4y = 36\] helps visualize the intercepts and the shape of the graph. To graph it, follow these steps:
  • Start by plotting the x-intercepts at (2, 0) and (-2, 0)
  • Next, plot the y-intercept at (0, 9)
  • Draw a smooth curve through these points, understanding that the quadratic nature implies a symmetric parabola opening upwards or sideways, depending on the coefficient values
This visual aid reinforces the algebraic solutions and helps understand the relationship between the equation and its graph.

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Most popular questions from this chapter

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the vertex of the graph of \(f(x)=3 x^{2}-12 x+7\)

Find \(g(3)\) where \(g(x)=\left\\{\begin{array}{ll}3 x^{2}-7 x & \text { if } \quad x<0 \\ 5 x-9 & \text { if } \quad x \geq 0\end{array}\right.\) Find \(g(3)\) where $$ g(x)=\left\\{\begin{array}{ll} 3 x^{2}-7 x & \text { if } \quad x<0 \\ 5 x-9 & \text { if } \quad x \geq 0 \end{array}\right. $$

Suppose \(f\) is a polynomial function. If \(f(-2)=7\) and \(f(6)=-1,\) then the Intermediate Value Theorem guarantees which of the following? Justify your answer. (a) \(f(0)=0\) (b) \(f(c)=3\) for at least one number \(c\) between -2 and 6 . (c) \(f(c)=0\) for at least one number between -1 and 7 . (d) \(-1 \leq f(x) \leq 7\) for all numbers in the closed interval [-2,6]

Graph each polynomial function. $$ f(x)=2 x^{3}-x^{2}+2 x-1 $$

Factor completely: \(6 x^{4} y^{4}+3 x^{3} y^{5}-18 x^{2} y^{6}\)
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