Problem 19 - Beginning and Intermediate Algebra

Question

For quadratic function, identify the vertex, axis of symmetry, and \(x\)- and \(y\)-intercepts. Then, graph the function. \(h(x)=\frac{1}{2}(x+4)^{2}\)

Step-by-Step Solution

Answer

The vertex of the quadratic function \(h(x)=\frac{1}{2}(x+4)^2\) is \((-4, 0)\). The axis of symmetry is the vertical line \(x=-4\). The x-intercept is the same as the vertex at \((-4, 0)\), and the y-intercept is at \((0, 8)\). To graph the function, plot the vertex, axis of symmetry, x- and y-intercepts, and a few additional points, then draw a parabola going through all the points, with the axis of symmetry being \(x = -4\).

Step 1: Find the Vertex

Since the given function is in the form \(f(x) = a(x-h)^2 + k\), we can easily identify the vertex. In this case, \(a = \frac{1}{2}, h = -4\), and \(k = 0\). Therefore, the vertex is \((h, k) = (-4, 0)\).

Step 2: Find the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex's x-coordinate. In this case, the vertex is \((-4, 0)\), so the axis of symmetry is the vertical line \(x = -4\).

Step 3: Find the x-intercepts

To find the x-intercepts, set \(h(x) = 0\) and solve for x: \[ \frac{1}{2}(x+4)^2 = 0 \] Since \((x+4)^2\) must be equal to zero for the whole expression to be zero, we have that \(x+4 = 0\), or \(x = -4\). So, the x-intercept is \((-4, 0)\), which is also the vertex in this case.

Step 4: Find the y-intercepts

To find the y-intercepts, set \(x = 0\) and solve for y: \[ h(0) = \frac{1}{2}(0+4)^2 = \frac{1}{2}(4)^2 = (0,8) \] The y-intercept is \((0, 8)\).

Step 5: Graph the Function

Now, we'll graph the function using the information we've gathered: - Vertex: \((-4, 0)\) - Axis of Symmetry: \(x = -4\) - x-intercept: \((-4, 0)\) - y-intercept: \((0, 8)\) To facilitate the graphing, we can plot some more points by choosing x-values to the right or left of the vertex and find the corresponding y-values (For this case, you could take x=-3,-2,1). Then, we'll plot these points and draw a parabola that goes through all the points, with the axis of symmetry being \(x = -4\).

Key Concepts

Vertex of a Quadratic Function

A quadratic function has a specific point called the "vertex," which is a type of optimal point for the parabola. This vertex represents either the maximum or the minimum point of the function, depending on whether the parabola opens upwards or downwards. For the function given by \[h(x) = \frac{1}{2}(x+4)^2\]you can immediately identify the vertex by the form \(f(x) = a(x-h)^2 + k\). Here:- \(a = \frac{1}{2}\)- \(h = -4\)- \(k = 0\)Thus, the vertex is \((-4, 0)\). When plotting the graph, this point acts as a pivot around which the parabola forms. It also provides valuable insight into the function's general shape and orientation, because the parabola will be centered around this point and will open upwards as \(a > 0\).

Axis of Symmetry

Every parabola has a line called the "axis of symmetry." This is a vertical line that divides the parabola into two mirrored halves. It passes through the vertex of the quadratic function, ensuring that each side of the parabola is a mirror image of the other. Knowing the axis of symmetry is useful when graphing because it shows the center line around which the curve is drawn.For the function \[h(x) = \frac{1}{2}(x+4)^2\]the vertex is \((-4, 0)\). This makes the axis of symmetry the line \(x = -4\). On the graph, you'll notice that for every point on the parabola on one side of this line, there's a corresponding point directly opposite on the other side.

x-intercept and y-intercept

Identifying intercepts is key to sketching the shape and position of a quadratic function. The points where the curve crosses the axes provide distinct markers.**X-Intercept**The x-intercept occurs where the graph crosses the x-axis. To find it, set the function equal to zero and solve for \(x\). For \[ \frac{1}{2}(x+4)^2 = 0 \], solving gives \(x = -4\).Therefore, the x-intercept is \((-4, 0)\), which is exactly at the vertex.**Y-Intercept**The y-intercept happens where the graph crosses the y-axis. This is found by evaluating the function at \(x = 0\): \[h(0) = \frac{1}{2}(0+4)^2 = 8\].Hence, the y-intercept is located at \((0, 8)\). These intercepts provide points that you can plot to get a shape of the curve easily.

Graphing Quadratic Functions

Graphing quadratic functions allows for a visual understanding of the curve's behavior. Through plotting key points like the vertex and intercepts, you begin forming the basis of the graph.For the quadratic function \[h(x) = \frac{1}{2}(x+4)^2\], follow these steps:- **Plot the vertex**: Start with \((-4, 0)\) because it's the central point from where your graph will extend.- **Draw the axis of symmetry**: Sketch the vertical line \(x = -4\), making sure each side of the graph is balanced.- **Add intercepts**: Both \((-4, 0)\) and \((0, 8)\) offer crucial reference points.To enrich your graph further, select additional x-values on either side of the axis of symmetry to calculate corresponding y-values. Plot these additional points. Connect all these plotted points with a smooth curve that rises on both ends if \(a > 0\) (which it does here). This visualization helps cement your understanding of how the quadratic function behaves across different x-values.