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Chapter 10: Problem 178

In the following exercises, find the \(x\) - and \(y\) -intercepts. $$ y=-x^{2}-14 x-49 $$

Short Answer

Expert verified
The y-intercept is \( (0, -49) \) and the x-intercepts are \( (-7, 0) \).

Step by step solution

01

Identify the equation

The given equation is: \[ y = -x^{2} - 14x - 49 \]
02

Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation and solve for \( y \):\[y = -(0)^2 - 14(0) - 49 \]\[ y = -49 \]So, the y-intercept is \( (0, -49) \).
03

Find the x-intercepts

To find the x-intercepts, set \( y = 0 \) in the equation and solve for \( x \):\[ 0 = -x^{2} - 14x - 49 \]Solve the quadratic equation \[ x^{2} + 14x + 49 = 0 \]Notice this is a perfect square trinomial:\[ (x + 7)^2 = 0 \]So, \( x + 7 = 0 \)Thus, \( x = -7 \).So, the x-intercepts are \( (-7, 0) \).
04

Summarize the intercepts

The y-intercept is \( (0, -49) \) and the x-intercepts are \( (-7, 0) \).

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Key Concepts

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

quadratic equations
A quadratic equation is a type of polynomial equation of the form:

\[ ax^2 + bx + c = 0 \]

Here,
  • a, b, and c are constants where a eq 0
  • x is the variable
Quadratic equations are also known as second-degree polynomials because the highest power of the variable is 2.

The graph of a quadratic equation is a parabola that opens upwards if a > 0 and downwards if a < 0.

The solutions to a quadratic equation, also known as the roots, can be found using various methods, such as factoring, completing the square, or the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
finding intercepts
Intercepts are points where a graph intersects the axes. They are key in graphing and understanding equations:

  • Y-intercept: This is the point where the graph crosses the y-axis. To find the y-intercept, set x = 0 in the equation and solve for y. For the given equation y = -x^2 - 14x - 49, when x = 0, we get y = -49. So, the y-intercept is (0, -49).

  • X-intercepts: These are points where the graph intersects the x-axis. To find the x-intercepts, set y = 0 in the equation and solve for x. For the equation 0 = -x^2 - 14x - 49, we solve the quadratic equation to get x = -7. So, the x-intercepts are (-7, 0).

Understanding intercepts helps in sketching the graph and analyzing the behavior of the function.
solving equations
Solving equations involves finding the values of the variable that make the equation true. For quadratic equations, there are several methods:

  • Factoring: This method involves expressing the quadratic equation as a product of its factors. If a quadratic equation can be factored completely, then the solutions can be found by setting each factor to zero.

  • Quadratic Formula: The quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] solves any quadratic equation. It is derived from the standard form of a quadratic equation.

  • Completing the Square: This method involves rewriting the quadratic equation in such a way that one side becomes a perfect square trinomial. This makes it easier to solve for the roots.
In the given solution, factoring was used because it was evident that the equation \[ x^2 + 14x + 49 = 0 \] is a perfect square trinomial and could be simplified as \[ (x + 7)^2 = 0 \]. Solving that gives \[ x = -7 \].

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Most popular questions from this chapter

Solve by using the Quadratic Formula. \(4 m^{2}+m-3=0\)

In the following exercises, determine if the parabola opens up or down. $$ y=4 x^{2}+x-4 $$

In the following exercises, determine the number of solutions to each quadratic equation. a. \(25 p^{2}+10 p+1=0\) b.\(7 q^{2}-3 q-6=0\) c.\(7 y^{2}+2 y+8=0\) d.\(25 z^{2}-60 z+36=0\)

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry. $$ y=x^{2}+6 x+8 $$

Solve by completing the square. \(3 q^{2}-5 q=9\)
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