Problem 14 - Beginning and Intermediate Algebra

Question

For quadratic function, identify the vertex, axis of symmetry, and \(x\)- and \(y\)-intercepts. Then, graph the function. \(g(x)=-(x-3)^{2}+2\)

Step-by-Step Solution

Answer

The vertex of the given quadratic function \(g(x) = -(x-3)^2 + 2\) is (3, 2), with its axis of symmetry at x = 3. The y-intercept is at the point (0, -7), and the x-intercepts are at the points \((3 - \sqrt{2}, 0)\) and \((3 + \sqrt{2}, 0)\). To graph the function, plot the vertex, axis of symmetry, x-intercepts, and y-intercept.

Step 1: Find the vertex

The given quadratic function is in the vertex form: \(g(x) = a(x - h)^2 + k\), where (h, k) represents the vertex of the parabola. In our case, we have: - a = -1 - h = 3 - k = 2 Thus, the vertex of the quadratic function is at the point (h, k) = (3, 2).

Step 2: Determine the axis of symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. Its equation is given by \(x = h\), where h is the x-coordinate of the vertex. Using the x-coordinate of our vertex (3), we have: Axis of symmetry: x = 3

Step 3: Find the y-intercept

To find the y-intercept, we set x = 0 in the given equation and solve for g(x): \(g(0) = -(0 - 3)^2 + 2\) \(g(0) = -(-3)^2 + 2\) \(g(0) = -9 + 2\) \(g(0) = -7\) The y-intercept is at the point (0, -7).

Step 4: Find the x-intercepts

To find the x-intercepts, we set g(x) = 0 and solve for x: \(- (x - 3)^2 + 2 = 0\) \((x - 3)^2 = 2\) Now, we take the square root of the both sides: \(x - 3 = \pm\sqrt{2}\) Next, we solve for x by adding 3: \(x = 3 \pm\sqrt{2}\) The x-intercepts are at the points \((3 - \sqrt{2}, 0)\) and \((3 + \sqrt{2}, 0)\).

Step 5: Graph the function

Using all the information found in previous steps, we can graph the function. Our graph will have the following important points and characteristics: - Vertex: (3, 2) - Axis of symmetry: x = 3 - Y-intercept: (0, -7) - X-intercepts: \((3 - \sqrt{2}, 0)\) and \((3 + \sqrt{2}, 0)\) Now sketch the graph using these points, and you will have graphed the quadratic function \(g(x) = -(x - 3)^2 + 2\).

Key Concepts

Vertex Form

The vertex form of a quadratic function is a very useful way of expressing the equation of a parabola. It is given by the formula \( g(x) = a(x - h)^2 + k \). This specific form allows us to quickly identify the vertex of the parabola, which is the point \((h, k)\).
The vertex is a key feature of any parabola. It is the highest or lowest point on the graph, depending on the direction the parabola opens. In the given quadratic function \( g(x) = -(x-3)^2+2 \), the vertex is at \((3, 2)\).

• The parameter \( a \) determines the direction and width of the parabola. When \( a \) is negative, as in our example where \( a = -1 \), the parabola opens downward.
• The parameters \( h \) and \( k \) are taken directly from the equation. Here, \( h = 3 \) and \( k = 2 \), giving the vertex at \((3, 2)\).

Axis of Symmetry

The axis of symmetry is an important concept in quadratic functions. It is a vertical line that divides the parabola into two mirror-image halves. This line always passes through the vertex of the parabola, making it easy to determine.
For the quadratic function \( g(x) = -(x-3)^2+2 \), the axis of symmetry can be found by the equation \( x = h \). Here, since our vertex is at \((3, 2)\), the axis of symmetry is at \( x = 3 \).

• This line does not only provide a visual guide but also helps when plotting the quadratic function accurately.
• Using the axis of symmetry, we can reflect points across this line to better understand the shape and position of the parabola.

X-Intercepts

Finding the x-intercepts of a quadratic function helps us understand the points where the parabola crosses the x-axis. These are the real solutions of the quadratic equation when \( g(x) = 0 \).
To find the x-intercepts in the function \( g(x) = -(x-3)^2+2 \), we set \( g(x) = 0 \) and solve for \( x \):

• First, rearrange to get \((x-3)^2 = 2\).
• Next, take the square root of both sides : \( x - 3 = \pm\sqrt{2} \).
• Finally, solve for \( x \) to obtain \( x = 3 \pm \sqrt{2} \).

Thus, the x-intercepts are roughly \((3 - \sqrt{2}, 0)\) and \((3 + \sqrt{2}, 0)\). These points tell us when the output of the quadratic equals zero, which is crucial for graphing.

Y-Intercepts

The y-intercept is another critical point of a quadratic function. It represents the point where the graph crosses the y-axis, which occurs when the input \( x \) is zero.
To find the y-intercept of the given quadratic function \( g(x) = -(x-3)^2+2 \), substitute \( x = 0 \) into the equation:

• Calculate \( g(0) = -(0-3)^2+2 \).
• Which simplifies to \( g(0) = -9 + 2 \).
• Thus, the y-intercept is \( (0, -7) \).

This single point provides insight into where the parabola will pass through the y-axis and is another essential part of plotting the entire curve on a graph.