Problem 31 - Beginning and Intermediate Algebra

Question

For each equation, identify the vertex, axis of symmetry, and \(x\) - and \(y\) -intercepts. Then, graph the equation. $$x=-2(y-2)^{2}-9$$

Step-by-Step Solution

Answer

The vertex of the parabola is at \((-9, 2)\), and the axis of symmetry is the vertical line given by the equation \(y = 2\). The x-intercept is at \((-17, 0)\), and the y-intercept is at \((0, 2)\). To graph the equation, plot the vertex, x-intercept, y-intercept, and draw a vertical line to represent the axis of symmetry, then sketch the parabola's shape using the points and the axis of symmetry.

Step 1: Identifying the vertex and axis of symmetry

The given equation is in the form \(x = a(y - k)^2 + h\). In this case, our \(h\) is -9 and our \(k\) is 2. Therefore, the vertex of the parabola is at the point \((-9, 2)\) and the axis of symmetry is a vertical line that passes through the vertex, which is given by the equation \(y = 2\).

Step 2: Finding the x-intercepts

To find the x-intercepts, we need to set \(y = 0\) and solve for \(x\). Plugging in \(y = 0\) into the given equation, we have: \(x = - 2(0 - 2)^2 - 9\) \(x = - 2(-2)^2 - 9\) \(x = -8 - 9\) \(x = -17\) So, there is one x-intercept is at \((-17, 0)\).

Step 3: Finding the y-intercepts

To find the y-intercepts, we need to set \(x = 0\) and solve for \(y\). Plugging in \(x = 0\) into the given equation, we get: \(0 = -2(y - 2)^2 - 9\) Now, we move the -9 to the other side of the equation and then divide by -2: \((-9 + 9)/2 = (y - 2)^2\) \(0 = (y - 2)^2\) Taking the square root of both sides, we obtain: \(y - 2 = 0\) \(y = 2\) So, the y-intercept is at \((0, 2)\).

Step 4: Graphing the equation

Now that we have identified the vertex, axis of symmetry, and x- and y-intercepts, we can graph the equation. Plot the vertex at \((-9, 2)\), the x-intercept at \((-17, 0)\), and the y-intercept at \((0, 2)\). Since the parabola is symmetric with respect to the line \(y = 2\), draw a vertical line at \(y = 2\) to represent the axis of symmetry. Finally, sketch the parabola's shape using the points and the axis of symmetry. The resulting graph represents the equation \(x = -2(y - 2)^2 - 9\).

Key Concepts

Vertex

The vertex of a parabola is a special point that gives us useful information about the shape and location of the parabola on a graph.
The vertex is either the highest or lowest point on the curve, depending on whether the parabola opens upwards or downwards.
For our specific equation, which is given in the form \( x = a(y - k)^2 + h \), the vertex can be directly extracted as the point \(h, k\).

• In our example, the vertex is located at \((-9, 2)\).
• It reflects where the curve changes direction.
  

The vertex is crucial, as it anchors the entire parabola and guides us in drawing the graph accurately. By knowing the vertex, we can tell a lot about the behavior of the parabola just by looking at its equation.

Axis of Symmetry

The axis of symmetry in a parabola is an imaginary vertical line that passes directly through the vertex, splitting the parabola into two equal halves.Because of this symmetry, the parabola will look the same on both sides of this line, making it much easier to graph.
For our equation, because it is presented in the form \( x = a(y - k)^2 + h \), the axis of symmetry is represented by the equation \( y = k \).

• In our example, the axis of symmetry is \( y = 2 \).
• This line passes through the vertex \((-9, 2)\).
  

Knowing the axis of symmetry helps in understanding how the parabola will behave along the y-direction and helps maintain accuracy when plotting the graph.

x-intercepts

The \(x\)-intercepts are points where the graph of the parabola crosses the x-axis. These points are significant because they tell us where the output value of the function is zero.
To find the \(x\)-intercepts, we need to set \(y = 0\) in the parabola's equation and solve for \(x\).
For our equation \( x = -2(y - 2)^2 - 9 \), it simplifies to:

• \( x = -2(0 - 2)^2 - 9 \)
• \( x = -8 - 9 = -17 \)
  

Thus, there is one \(x\)-intercept for our parabola at the point \((-17, 0)\). This means the parabola touches the x-axis only once, giving essential clues about the graph's shape.

y-intercepts

The \(y\)-intercepts are points where the graph of the parabola crosses the y-axis. Determining these points involves setting \(x = 0\) and solving for \(y\) in the equation. These intercepts are valuable because they show where the function's input is zero.
For our example, substituting \(x = 0\) into \(x = -2(y - 2)^2 - 9\) yields:

• \( 0 = -2(y - 2)^2 - 9 \)
• Simplifying gives \( (y - 2)^2 = 0 \)
• This results in \( y = 2 \)
  

Therefore, the \(y\)-intercept is at the point \((0, 2)\).These intercepts provide a starting point when sketching the parabola, helping ensure the curve is accurately aligned with the axis. They guide us in plotting the curve relative to other features like the vertex and axis of symmetry.