Problem 13 - Beginning and Intermediate Algebra

Question

For quadratic function, identify the vertex, axis of symmetry, and \(x\)- and \(y\)-intercepts. Then, graph the function. \(f(x)=-(x+3)^{2}+6\)

Step-by-Step Solution

Answer

The vertex of the quadratic function \(f(x) = -(x+3)^2 + 6\) is \((-3, 6)\), the axis of symmetry is \(x = -3\), the x-intercepts are \((-3+\sqrt{6}, 0)\) and \((-3-\sqrt{6}, 0)\), and the y-intercept is \((0, -3)\). To graph the function, plot the vertex, axis of symmetry, x-intercepts, and y-intercept, and then connect the points with a smooth curve making sure the parabola is opening downwards.

Step 1: Identify the Vertex

First, we need to identify the vertex of the parabola. This can be found by looking at the given quadratic function, \(f(x) = -(x+3)^2 + 6\). We can see that the vertex of the parabola is given by the coordinates (h, k), where h = -3 and k = 6. Therefore, the vertex of the parabola is (-3, 6).

Step 2: Identify the Axis of Symmetry

Next, we will find the axis of symmetry of the parabola. This is a vertical line that passes through the vertex and splits the parabola into two symmetrical halves. Since the vertex is at (-3, 6), the axis of symmetry is given by the equation \(x = -3\).

Step 3: Find the x-intercepts

To find the x-intercepts, we need to set f(x) to 0 and solve for x. We have: \(0 = -(x+3)^2 + 6\) Now, we simply solve for x: \((x+3)^2 = 6\) \(\sqrt{(x+3)^2} = \sqrt{6}\) \(x+3 = \pm\sqrt{6}\) \(x = -3 \pm\sqrt{6}\) So, the x-intercepts are \((-3+\sqrt{6}, 0)\) and \((-3-\sqrt{6}, 0)\).

Step 4: Find the y-intercept

To find the y-intercept, we need to set x to 0 and solve for f(x). We have: \(f(0) = - (0+3)^2 + 6\) \(f(0) = -(3)^2 + 6\) \(f(0) = -9 + 6\) \(f(0) = -3\) So, the y-intercept is \((0, -3)\).

Step 5: Graph the Function

Now that we have the vertex, axis of symmetry, x-intercepts, and y-intercept, we can graph the function: 1. Plot the vertex (-3,6) on the coordinate plane. 2. Draw the axis of symmetry, which is the vertical line \(x=-3\). 3. Plot the x-intercepts \((-3+\sqrt{6}, 0)\) and \((-3-\sqrt{6}, 0)\) and the y-intercept \((0, -3)\). 4. Connect the points with a smooth curve, making sure the parabola is opening downwards (since the coefficient of \((x+3)^2\) is negative). Now you have successfully graphed the quadratic function \(f(x) = -(x+3)^2 + 6\) and identified its key components.

Key Concepts

Vertex of a Parabola

In the equation of a quadratic function, represented as \[f(x) = a(x-h)^2 + k\], where \(a\), \(h\), and \(k\) are constants, the vertex of the parabola is a crucial point. It is denoted by the coordinates \((h, k)\). This point represents the peak or the lowest point (depending on whether the parabola opens upwards or downwards) of the parabola.
For the function \( f(x) = -(x+3)^2 + 6 \), the expression is in the standard form. Here, \(h = -3\) and \(k = 6\). Thus, the vertex is at the point \((-3, 6)\). The vertex helps us understand the direction and the highest point in the parabola since the \(a\) value is negative, indicating a downwards opening.

Axis of Symmetry

The axis of symmetry in a parabola is a vertical line that passes through the vertex and divides the parabola into two mirroring halves. It effectively acts like a mirror, ensuring each side of the parabola is identical to the other.
To find it, you simply look at the \(h\) value in the vertex form \((h, k)\) of the quadratic function. For example, in our given function \(f(x) = -(x+3)^2 + 6\), the axis of symmetry comes from the vertex \(h=-3\), hence it is \(x = -3\).
This line doesn't just help in sketching graphs; it also aids in predicting the behavior of the quadratic function. Each \(y\) value to the left of this line has an equal \(y\) value to its right at the same distance from the axis.

X-intercepts

The \(x\)-intercepts of a quadratic function are the points where the graph crosses the \(x\)-axis. At these points, the \(y\)-value is zero. To find them, set \(f(x)\) to zero and solve for \(x\).
For instance, with the function \(f(x) = -(x+3)^2 + 6\):

• \(0 = -(x+3)^2 + 6\)
• \((x+3)^2 = 6\)
• \(x+3 = \pm\sqrt{6}\)
• \(x = -3 \pm\sqrt{6}\)

Therefore, the \(x\)-intercepts are \((-3 + \sqrt{6}, 0)\) and \((-3 - \sqrt{6}, 0)\). These intercepts help to understand where the parabola crosses the horizontal \(x\)-axis.

Y-intercepts

The \(y\)-intercept of a function is the point where the graph crosses the \(y\)-axis. For quadratic functions, you find it by setting \(x\) to zero and solving for \(f(x)\). This method offers a quick way to determine one key point on the graph.
Consider the equation \(f(x) = -(x+3)^2 + 6\). Substitute \(x = 0\):

• \(f(0) = -(0+3)^2 + 6\)
• \(f(0) = -9 + 6\)
• \(f(0) = -3\)

Thus, the \(y\)-intercept is \((0, -3)\). This intercept provides a concrete point through which the graph passes, helping in sketching the curve and understanding its positioning on the coordinate plane.