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Chapter 9: Problem 251

Find the intercepts of the parabola whose function is given. $$f(x)=-x^{2}-6 x-9$$

Short Answer

Expert verified
The y-intercept is (0, -9) and the x-intercept is (-3, 0).

Step by step solution

01

- Find the y-intercept

To find the y-intercept of the function, set x to 0 and solve for f(x). Substitute x = 0 into the function: \[ f(0) = -0^2 - 6(0) - 9 = -9 \] Thus, the y-intercept is the point (0, -9).
02

- Find the x-intercepts

To find the x-intercepts, set f(x) to 0 and solve for x. The function given is: \[ f(x) = -x^2 - 6x - 9 \]We set f(x) = 0: \[ -x^2 - 6x - 9 = 0 \] Multiply the entire equation by -1 to make the leading coefficient positive: \[ x^2 + 6x + 9 = 0 \]Now, factor the quadratic equation: \[ (x + 3)(x + 3) = 0 \]This can be written as: \[ (x + 3)^2 = 0 \] Solve for x by taking the square root of both sides: \[ x + 3 = 0 \]Therefore, \[ x = -3 \]So, the x-intercept is the point (-3, 0).

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Key Concepts

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

y-intercept
Finding the y-intercept of a parabola is quite straightforward. The y-intercept is where the parabola touches or crosses the y-axis. To determine this, we set the value of x to 0 and solve for y. In our equation:

either looking at the given function: 
  f(x) = -x^2 - 6x - 9
we substitute x = 0:n
 f(0) = -0^2 - 6(0) - 9 = -9
So, the y-intercept is the point (0, -9). This is because when x is 0, f(x) equals -9.

Quick Tip: The y-intercept is always at the point where x = 0. Simply substitute 0 for x and solve the remaining equation for y.
x-intercepts
The x-intercepts of a parabola are points where the parabola crosses the x-axis. To find these points, we need to solve the equation for when f(x) = 0. For the given function: 

  f(x) = -x^2 - 6x - 9
By setting f(x) to 0, we get:
  0 = -x^2 - 6x - 9
To simplify solving, multiply the entire equation by -1 to make the leading coefficient positive:
 x^2 + 6x + 9 = 0
Next, we factor the quadratic equation to find its x-intercepts.

Tip: x-intercepts are where y equals 0. So set the function to 0 and solve for x.
quadratic equation
A quadratic equation is generally in the form of: 
 ax^2 + bx + c
where a, b, and c are coefficients. For our specific equation: 

  f(x) = -x^2 - 6x - 9,
Here, a = -1, b = -6, and c = -9. Quadratic equations form parabolas when graphed. The coefficient 'a' determines whether the parabola opens upwards (when a > 0) or downwards (when a < 0). Since our 'a' is -1, the parabola opens downwards.

Important Note: When solving for x-intercepts, we first set the quadratic equation to zero and then solve for x using factoring or other methods like the quadratic formula.
factoring quadratics
Factoring is a method used to solve quadratic equations. In our case: 

  x^2 + 6x + 9 = 0
We look for two numbers that multiply to 9 and add to 6. In this situation, both numbers are 3. Therefore, we can rewrite the equation as: 
  (x + 3)(x + 3) = 0
So it can be written as: 
  (x + 3)^2 = 0,
To solve for x, take the square root of both sides: 
  x + 3 = 0,
Thus, 
  x = -3.
So, the x-intercept is at (-3, 0).

Recap: Factoring quadratics involves finding factors of c that add up to b and rewriting the equation using these factors.

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Most popular questions from this chapter

Graph each function using transformations. $$f(x)=(x+2)^{2}+1$$

Find the intercepts of the parabola whose function is given. $$f(x)=x^{2}+8 x+12$$

Solve. Round answers to the nearest tenth. A retailer who sells backpacks estimates that by selling them for \(x\) dollars each, he will be able to sell \(100-x\) backpacks a month. The quadratic function \(R(x)\) \(=-x^{2}+100 x\) is used to find the \(R,\) received when the selling price of a backpack is \(x .\) Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

Solve. Round answers to the nearest tenth. A cell phone company estimates that by charging \(x\) dollars each for a certain cell phone, they can sell \(8-\) \(x\) cell phones per day. Use the quadratic function \(R(x)=\) \(-x^{2}+8 x\) to find the revenue received per day when the selling price of a cell phone is \(x .\) Find the selling price that will give them the maximum revenue per day, and then find the amount of the maximum revenue.

Find the maximum or minimum value of each function. $$y=-9 x^{2}+16$$
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