Question

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

Short answer

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

Step-by-step solution

Identify the intercepts

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

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: \[9 x^{2} + 4 \times 0 = 36\] \[9 x^{2} = 36\] \[x^{2} = 4\] \[x = \text{±2}\] The x-intercepts are at (2, 0) and (-2,0).

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).

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).

Key concepts

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 \[x = \text{±2}\]. 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., \[x = \text{±}\frac{\text{Root of k}}{a}\] 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.