Problem 17 - Algebra A Combined Function

Question

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and graph the function. $$ f(x)=x^{2}-4 $$

Step-by-Step Solution

Answer

The vertex is \((0, -4)\), the parabola opens upward, \(x\)-intercepts are \((-2,0)\) and \((2,0)\), \(y\)-intercept is \((0, -4)\).

Step 1: Identify the Standard Form

The given quadratic function is \(f(x) = x^2 - 4\). This is already in the standard form \(f(x) = ax^2 + bx + c\), where \(a = 1\), \(b = 0\), and \(c = -4\).

Step 2: Determine the Direction of the Parabola

The coefficient \(a = 1\) is positive. Therefore, the parabola opens upward.

Step 3: Find the Vertex

The vertex of a parabola in standard form \(ax^2 + bx + c\) is given by the formula \(x = -\frac{b}{2a}\). Substituting the values of \(a\) and \(b\), we get \(x = -\frac{0}{2(1)} = 0\). To find the \(y\)-value of the vertex, substitute \(x = 0\) back into the original equation: \(f(0) = 0^2 - 4 = -4\). Thus, the vertex is \((0, -4)\).

Step 4: Calculate the Intercepts

For the \(y\)-intercept, substitute \(x = 0\) into the function: \(f(0) = -4\). Thus, the \(y\)-intercept is \((0, -4)\).To find the \(x\)-intercepts, set \(f(x) = 0\) and solve for \(x\): \((x^2 - 4 = 0)\). This factors to \((x-2)(x+2)=0\), giving \(x = 2\) and \(x = -2\). Hence, the \(x\)-intercepts are \((2, 0)\) and \((-2, 0)\).

Step 5: Sketch the Graph

To graph \(f(x) = x^2 - 4\), plot the vertex at \((0, -4)\), \(y\)-intercept at \((0, -4)\), and \(x\)-intercepts at \((2, 0)\) and \((-2, 0)\). Draw a smooth parabola opening upwards passing through these points.

Key Concepts

Vertex Determination

The vertex of a quadratic function is a crucial point on its graph. It's the highest or lowest point, depending on the direction the parabola opens. When the quadratic function is in the standard form \( ax^2 + bx +c \), you determine the vertex using the formula \( x = -\frac{b}{2a} \).

For the function \( f(x) = x^2 - 4 \), the coefficients are \( a = 1 \), \( b = 0 \), and \( c = -4 \). Plug these values into the formula:

• \( x = -\frac{0}{2 \times 1} = 0 \)

To find the \( y \)-coordinate of the vertex, substitute \( x = 0 \) back into the equation to get \( f(0) = 0^2 - 4 = -4 \).

This means the vertex is located at \((0, -4)\). This point is essential for sketching the graph.

Parabola Direction

The direction in which a parabola opens is determined by the coefficient \( a \) in the standard form \( ax^2 + bx + c \). If \( a > 0 \), the parabola opens upward. If \( a < 0 \), it opens downward.

In our quadratic function, \( f(x) = x^2 - 4 \), we have \( a = 1 \), which is a positive number.

• This tells us that the parabola opens upward.

This direction means the graph will have a minimum point at the vertex and will extend infinitely upwards. Knowing the direction is necessary when graphing the function, ensuring that the shape accurately reflects the equation. By identifying whether a parabola is opening upwards or downwards, you can more easily spot maximum or minimum value scenarios in a real-world context.

Intercepts Calculation

To graph a quadratic function completely, you'll need to determine where it intersects the axes: the intercepts.
**Calculating the \( y \)-intercept:**

• To find the \( y \)-intercept, set \( x = 0 \) in the function. For \( f(x) = x^2 - 4 \), this is simply \( f(0) = 0^2 - 4 = -4 \).
• The \( y \)-intercept is at the point \((0, -4)\).

**Calculating the \( x \)-intercepts:**

• Set the function equal to zero: \( x^2 - 4 = 0 \).
• Factorize the equation: \((x - 2)(x + 2) = 0\).
• Solving gives \( x = 2 \) and \( x = -2 \).
• The \( x \)-intercepts are the points \((2, 0)\) and \((-2, 0)\).

Each intercept is a point where the graph intersects an axis, providing crucial reference points for sketching the parabola.

Graph Sketching

Sketching the graph of a quadratic function involves plotting key points and drawing the correct shape of the curve. Start by marking the vertex on the coordinate plane. For \( f(x) = x^2 - 4 \), plot the vertex at \((0, -4)\).

Then, add the \( x \)-intercepts at \((2, 0)\) and \((-2, 0)\), and the \( y \)-intercept again at \((0, -4)\). These points give a framework for the graph.

• Connect these points with a smooth, curved line reflecting the direction instructed by \( a \).
• In this case, ensure the parabola opens upwards from the vertex.

The resulting curve should pass through all plotted points, expanding outward as it moves away from the vertex. This visual representation provides a clear picture of how the function behaves across different inputs. Remember, drawing by hand or in a digital tool takes some practice, but with these steps, you can accurately depict the function's graph.