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Chapter 8: Problem 46

Find any \(x\) -intercepts and the \(y\) -intercept. If no \(x\) -intercepts exist, state this. $$g(x)=x^{2}-6 x+9$$

Short Answer

Expert verified
x-intercept: (3,0); y-intercept: (0,9)

Step by step solution

01

- Determine the x-intercepts

To find the x-intercepts, set the function equal to zero and solve for x. g(x) = x^2 - 6x + 90 = x^2 - 6x + 9Factor the quadratic equation: 0 = (x-3)^2Solve for x: (x-3)^2 = 0x - 3 = 0x = 3Thus, the x-intercept is at (3,0).
02

- Determine the y-intercept

To find the y-intercept, set x equal to 0 and solve for y. g(0) = 0^2 - 6(0) + 9g(0) = 9Thus, the y-intercept is at (0,9).
03

- State the intercepts

The x-intercept is at (3, 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-intercept
To understand x-intercepts in a quadratic function, let's delve into the details step by step. An x-intercept is where the graph of the function crosses the x-axis. This happens when the output of the function is zero. To find the x-intercept of the given quadratic function:
  1. Set the quadratic function equal to zero. For instance, if the function is given by \(g(x) = x^2 - 6x + 9\), set it as \(0 = x^2 - 6x + 9\).
  2. Simplify or factorize the equation. In our case, \(x^2 - 6x + 9\) can be factored as \((x-3)^2\).
  3. Solve for \(x\). Here, we get \((x-3)^2 = 0\). Thus, \(x = 3\).
So the x-intercept for this quadratic function is at the point \((3,0)\).Exploring how to find x-intercepts using this basic process helps you to understand how quadratic functions behave and where they cross the x-axis.
y-intercept
Now let's talk about the y-intercept of a quadratic function. The y-intercept is where the graph crosses the y-axis. This happens when the input (or \(x\)) is zero. Let's understand this with our example function \(g(x) = x^2 - 6x + 9\):
  • To find the y-intercept, set \(x = 0\) in the function.
  • Substitute x with 0 into the equation: \(g(0) = 0^2 - 6(0) + 9\).
  • Simplify the equation. Here, \(g(0) = 9\).
Hence, the y-intercept for this quadratic function is at \((0,9)\).Understanding the y-intercept helps you see where the function crosses the y-axis, providing another anchor point for the graph of the quadratic function.
quadratic function
Quadratic functions are one of the fundamental structures in algebra and can be represented in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. These functions graph into a shape called a parabola. Key features of quadratic functions include:
  • The highest degree of the variable \(x\) is 2.
  • They can open upwards or downwards depending on the sign of \(a\). If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.
  • The vertex is the highest or lowest point on the graph depending on the direction in which the parabola opens.
For our example \(g(x) = x^2 - 6x + 9\), the quadratic function can additionally be expressed in its factored form as \((x-3)^2\), indicating the parabola touches the x-axis at \((3,0)\), its vertex. Familiarizing yourself with the various representations of quadratic functions aids in graphing and solving them efficiently.
solving equations
Solving equations, especially quadratic ones, is a critical skill in math. For quadratic equations of the form \(ax^2 + bx + c = 0\), there are several methods to find the solutions (or roots), which are the x-intercepts in the context of graphs. Let's highlight some common methods:
  • Factoring: Simplify the quadratic expression into products of binomials and solve for the variable. Like in our example where \(x^2 - 6x + 9\) becomes \((x-3)^2=0\).
  • Quadratic Formula: Use the formula \(x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a}\) to find the roots directly.
  • Completing the Square: Rearrange the equation into a perfect square trinomial form and solve for the variable.
  • Graphing: Visually identify the points where the quadratic curve intersects the x-axis.
Understanding these different methods can help you tackle a variety of quadratic equations efficiently. Each method has its own ideal situations, and mastering them will give you flexibility in solving quadratic equations.

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