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Chapter 7: Problem 9

Sketch the graph of each parabola by using the vertex, the \(y\) -intercept, and the \(x\) -intercepts. Check the graph using a calculator. $$y=x^{2}-4$$

Short Answer

Expert verified
The parabola has a vertex at (0, -4), y-intercept at (0, -4), and x-intercepts at (2, 0) and (-2, 0).

Step by step solution

01

Identify the Vertex

The vertex of a parabola in the form \( y = a(x-h)^2 + k \) is \((h, k)\). For the equation \( y = x^2 - 4 \), it can be rewritten as \( y = (x-0)^2 - 4 \). Thus, the vertex is \((0, -4)\).
02

Find the y-intercept

The \( y \)-intercept is the point where the graph intersects the \( y \)-axis. Set \( x = 0 \) in the equation \( y = x^2 - 4 \). This gives \( y = 0^2 - 4 = -4 \). Thus, the \( y \)-intercept is \((0, -4)\).
03

Find the x-intercepts

The \( x \)-intercepts are the points where the graph intersects the \( x \)-axis, which occur when \( y = 0 \). Set \( y = 0 \) in the equation to solve for \( x \). \( 0 = x^2 - 4 \) simplifies to \( x^2 = 4 \). Taking the square root of both sides gives \( x = \pm 2 \). Therefore, the \( x \)-intercepts are \((2, 0)\) and \((-2, 0)\).
04

Sketch the Graph

Plot the vertex \((0, -4)\), the \( y \)-intercept \((0, -4)\), and the \( x \)-intercepts \((2, 0)\) and \((-2, 0)\) on the coordinate plane. Since the parabola opens upwards (it has the standard \( y = x^2 \) form), draw a smooth curve passing through these points, forming a U-shaped graph.
05

Verify with a Calculator

Use a graphing calculator to input \( y = x^2 - 4 \) and observe the graph. It should match your sketch with the vertex at \((0, -4)\), the \( y \)-intercept at \((0, -4)\), and the \( x \)-intercepts at \((2, 0)\) and \((-2, 0)\). This confirms the parabola is correctly sketched.

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Key Concepts

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

Understanding the Vertex of a Parabola
The vertex of a parabola is like the tip of an iceberg, the point that anchors the entire structure. For parabolas in the form \(y = a(x-h)^2 + k\), the vertex is conveniently located at the point \((h, k)\). Understanding this helps us determine where the parabola will turn or shift direction.
When you have the equation \(y = x^2 - 4\), it can be seen as \(y = (x-0)^2 - 4\). Here, the vertex is \((0, -4)\).
  • The \(h\) value is \(0\), which means there is no horizontal shift, keeping the parabola centered on the y-axis.
  • The \(k\) value is \(-4\), pushing our vertex down 4 units below the origin.

Grasping the concept of the vertex helps you immediately locate the lowest point of the parabola (for \(a > 0\)), supporting all further steps in graphing.
Finding the X-intercepts
X-intercepts are vital as they reveal where the parabola crosses the x-axis. These intercepts represent the real roots of the quadratic equation and provide essential reference points for sketching. To find your x-intercepts, solve the quadratic equation where \(y = 0\). For \(y = x^2 - 4\), set it to zero and find:
\[ x^2 - 4 = 0 \]
  • Adding 4 to both sides gives us \(x^2 = 4\).
  • Taking the square root from both sides, we find \(x = \pm 2\).

Thus, the x-intercepts are \((2, 0)\) and \((-2, 0)\). This tells us the parabola crosses the x-axis twice, reflecting its symmetrical nature.
  • The points are equidistant from the vertex, which is characteristic of parabolas, giving a balanced, U-shaped curve.
Determining the Y-intercept
The y-intercept of a parabola is the point at which the parabola crosses the y-axis. Finding this point involves substituting \(x = 0\) in the equation, as the y-axis itself is characterized by an \(x\) value of zero.
For the equation \(y = x^2 - 4\), plugging in \(x = 0\) results in:
\[ y = 0^2 - 4 = -4 \]
Thus, the y-intercept is \((0, -4)\). This point actually coincides with the vertex in this example, offering a glimpse of symmetry in the graph.
  • Because the vertex and y-intercept align, it streamlines the graphing process.
  • It's essential to recognize this point immediately reveals how far vertically the parabola has shifted from the origin.

Understanding the y-intercept's location serves as an anchor when plotting or verifying your graph, ensuring its accuracy and alignment with expected quadratic behavior.

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