Problem 12 - Algebra A Combined Function

Question

Sketch the graph of each function. $$ f(x)=x^{2}-4 $$

Step-by-Step Solution

Answer

The graph is a parabola opening upwards with vertex at (0, -4), and it intersects the x-axis at (-2, 0) and (2, 0).

Step 1: Identify the Type of Function

The given function is \(f(x) = x^2 - 4\), which is a quadratic function. This type of function forms a parabola when graphed.

Step 2: Determine the Vertex

The standard form of a quadratic function is \(f(x) = ax^2 + bx + c\). For \(f(x) = x^2 - 4\), we have \(a = 1\), \(b = 0\), and \(c = -4\). The vertex of a parabola in standard form \(f(x) = ax^2 + bx + c\) is given by \(\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)\). Here, \(-\frac{0}{2 \cdot 1} = 0\), so the vertex is at \( (0, -4) \).

Step 3: Determine the Axis of Symmetry

The axis of symmetry for the parabola is the vertical line that passes through the vertex of the parabola. Since the vertex is \((0, -4)\), the axis of symmetry is the line \(x = 0\).

Step 4: Find the Y-intercept

To find the y-intercept, substitute \(x=0\) into the function: \(f(0) = 0^2 - 4 = -4\). So, the y-intercept is \((0, -4)\).

Step 5: Find the X-intercepts

To find the x-intercepts, set \(f(x) = 0\) and solve for \(x\): \(x^2 - 4 = 0\). This can be factored as \((x + 2)(x - 2) = 0\). Thus, \(x = 2\) and \(x = -2\), which means the x-intercepts are \((2, 0)\) and \((-2, 0)\).

Step 6: Plot Key Points

On a coordinate plane, plot the vertex \((0, -4)\), the y-intercept \((0, -4)\), and the x-intercepts \((2, 0)\) and \((-2, 0)\). These points provide a guideline for the shape of the parabola.

Step 7: Draw the Parabola

Draw a smooth curve passing through the plotted points to form a U-shaped parabola that opens upwards, reflecting the fact that the coefficient of \(x^2\) is positive.

Key Concepts

Parabola

A quadratic function like the one given, \(f(x) = x^2 - 4\), always graphs as a parabola. A parabola is a U-shaped curve that can open upwards or downwards. In this case, it opens upwards because the coefficient of \(x^2\) is positive. Parabolas are symmetric, meaning they are mirror images across a line called the axis of symmetry.

• They have a unique point called the vertex.
• Two important aspects are the x-intercepts where the graph crosses the x-axis and the y-intercept where it crosses the y-axis.

Visualizing a parabola helps in understanding the shape and direction of quadratic functions.

Vertex

The vertex of a parabola is the peak or the lowest point, depending on its orientation. For the function \( f(x) = x^2 - 4 \), the vertex is found using the formula \((-\frac{b}{2a}, f(-\frac{b}{2a}))\). Given that \(b = 0\) and \(a = 1\), the vertex is at \((0, -4)\).

• It represents the minimum point of the parabola since it opens upward.
• Understanding the vertex's location helps determine the overall shape.
• The y-value of the vertex provides the minimum value of the function when it opens upwards.

This key point is crucial in sketching and analyzing the quadratic function.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For a function in the form \(f(x) = ax^2 + bx + c\), the axis of symmetry can be derived from the vertex as \(x = -\frac{b}{2a}\). In our example, this results in \(x = 0\).

• This line passes directly through the vertex.
• It is helpful for symmetry as it indicates every point on one side has a corresponding point directly across.

The x-coordinate of the axis of symmetry matches the x-coordinate of the vertex, guiding the graphing process and comprehending the quadratic function's behavior.

X-intercepts

X-intercepts are the points at which the graph crosses the x-axis. For the function \(f(x) = x^2 - 4\), we find the x-intercepts by solving \(x^2 - 4 = 0\). This can be factored into \((x + 2)(x - 2) = 0\), resulting in x-values of 2 and -2.

• The x-intercepts are thus (2, 0) and (-2, 0).
• These points are also known as roots or zeros of the function.

They are significant because they indicate where the parabola touches or crosses the x-axis, crucial for sketching the parabola accurately. Understanding these intercepts adds clarity to the parabola's relationship with the x-axis.