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Chapter 0: Problem 50

Find the \(x\) - and \(y\)-intercepts of the graph of the equation. \(y=(x+3)^2\)

Short Answer

Expert verified
The x-intercept of the equation is \(x = -3\), and the y-intercept is \(y = 9\).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \(y=0\) in the function and solve for \(x\). So \(0 = (x+3)^2\). To solve for \(x\), take the square root of both sides. We get \(x + 3 = 0\) or \(x + 3 = -0\). After simplifying, we find that \(x = -3\) or \(x = -3\). Therefore, the x-intercept is -3.
02

Find the y-intercept

To find the y-intercept, set \(x=0\) in the function and solve for \(y\). So \(y = (0 + 3)^2 = 9\). Therefore, the y-intercept is \(y = 9\).

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Key Concepts

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

x-intercept calculation
To find the x-intercept of a quadratic graph, you need to determine where the graph crosses the x-axis. At these points, the y-value is always zero. For any given function like our equation, which is \[ y = (x+3)^2 \],set \( y = 0 \) and solve for \( x \).

Following these steps:
  • Substitute \( y = 0 \), so the equation becomes \( (x+3)^2 = 0 \).
  • Take the square root of both sides: \( x+3 = 0 \).
  • Solve the resulting simple equation: \( x = -3 \).
Thus, the x-intercept of your function is at \( x = -3 \).

Remember, finding x-intercepts provides key points where your graphed equation will be at ground level on the graph, crossing the horizontal axis. This concept is essential for understanding how quadratic functions unfold across a Cartesian plane.
y-intercept calculation
The y-intercept of a graph is the point where it intersects the y-axis. At this point, the x-value is zero, making it easy to find by substituting \( x = 0 \) into the function. For our quadratic equation \[ y = (x+3)^2 \],follow these simple steps to find the y-intercept:

  • Set \( x = 0 \), resulting in \( y = (0+3)^2 \).
  • Simplify the expression: \( y = 3^2 = 9 \).
Consequently, the y-intercept is at \( y = 9 \).

Recognizing the y-intercept can greatly help in sketching the graph of an equation. It gives you a fixed point where the curve begins its journey across the varying x-values.
solving quadratic equations
Solving quadratic equations is a fundamental aspect of understanding parabolic graphs. These equations generally have the form \[ ax^2 + bx + c = 0 \].In our exercise, the equation simplifies as\[ y = (x+3)^2 \],which makes it easier to handle using simple algebraic steps.

Ways to solve include:
  • Factoring: Rearrange the equation and split into factors to solve for \( x \). This method works well when expressions neatly factor into binomials.
  • Completing the Square: Rewrite the original form so you can find square roots more easily.
  • Using the Quadratic Formula: Plug into \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] to find the possible x-values for more complex quadratics.
For problems like ours with existing perfect squares, use simple solutions like rewriting or solving directly for minimal calculations. Understanding these methods allows you to break down most quadratic scenarios effectively, which is useful for plotting precise curves on a graph.

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Most popular questions from this chapter

Average Cost The inventor of a new game believes that the variable cost for producing the game is \(\$ 0.95\) per unit and the fixed costs are \(\$ 6000\). The inventor sells each game for \(\$ 1.69\). Let \(x\) be the number of games sold. (a) The total cost for a business is the sum of the variable cost and the fixed costs. Write the total cost \(C\) as a function of the number of games sold. (b) Write the average cost per unit \(\bar{C}=C / x\) as a function of \(x\).

In Exercises 33-38, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$ g(x)=\frac{4-x}{6} $$

Use a graphing utility to graph each function. Write a paragraph describing any similarities and differences you observe among the graphs. (a) \(y=x\) (b) \(y=x^{2}\) (c) \(y=x^{3}\) (d) \(y=x^{4}\) (e) \(y=x^{5}\) (f) \(y=x^{6}\)

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$ g(s)=4 s^{2 / 3} $$

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ g(x)=\frac{x}{8} $$
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