Question

Sketch the graph of the function. \(f(x)=x^{2}-9\)

Short answer

The graph is a parabola that opens upwards, with a vertex at (0, -9), x-intercepts at points (-3, 0) and (3, 0), and a y-intercept at (0, -9).

Step-by-step solution

Determine the Shape of the Graph

Looking at the function \(f(x) = x^{2}-9\), it can be determined that this is a quadratic function, where the highest exponent on the variable \(x\) is 2. In general, graphs of quadratic functions are parabolic. Since the coefficient of \(x^{2}\) is positive, the parabola will open upwards.

Find the Vertex

The vertex form of a quadratic function is \(f(x) = a(x-h)^{2}+k\), where \((h, k)\) are the coordinates of the vertex. The given function can be rewritten as \(f(x) = (x-0)^{2}-9\). Therefore, the vertex is at the point \((0, -9)\). The parabola will therefore be a U shape centered at point \((0, -9)\).

Find the X-intercepts (Roots)

The x-intercepts of the function can be found by setting \(f(x)\) equal to 0 and solving for \(x\). So, you have \(0 = x^{2} - 9\). Solving this equation, you get \(x=3\) and \(x=-3\). Therefore, the graph intercepts the x-axis at the points \((-3, 0)\) and \((3, 0)\).

Find the Y-intercept

The y-intercept of the function can be found by setting \(x\) equal to 0 and solving for \(f(x)\). Doing so gives you \(f(0) = (0)^{2} - 9 = -9\). Therefore, the graph intercepts the y-axis at the point \((0, -9)\).

Sketch the Graph

Plot the points determined in the previous steps: the vertex at \((0, -9)\), the x-intercepts at \((-3, 0)\) and \((3, 0)\), and the y-intercept also at \((0, -9)\). Finally, connect these points with a smooth curve to complete the graph of \(f(x) = x^{2} - 9\).

Key concepts

Parabolic Shapes

When graphing quadratic functions, the resulting shape is always a parabola. Imagine a symmetric curve that either opens upward like a cup or downward like an upside down cup, depending on the sign of the leading coefficient. In our example, the function is

f(x)=x^{2}-9. Since the coefficient of x^{2} is positive, the graph is a parabola that opens upwards. This is important to visualize as it gives us the general shape and direction of the graph before plotting specific points. Moreover, the width of the parabola depends on the absolute value of the coefficient of x^{2}. A larger coefficient causes a steeper, narrower parabola, while a smaller coefficient results in a wider one.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point, which is the 'tip' of the U or the inverted U shape. For the graph of

f(x)=x^{2}-9, the vertex is found by converting the equation into vertex form, which shows us directly where this crucial point lies. In this case, the given quadratic is already in a form that reveals the vertex coordinates, so f(x) can be interpreted as f(x) = (x-0)^{2}-9, indicating the vertex at (0, -9). Understanding the vertex helps us to graph the parabola accurately and easily find the axis of symmetry for the graph.

X-Intercepts

The x-intercepts, also known as roots or zeros, are the points where the graph crosses the x-axis. Finding these points is crucial for sketching the graph. To find x-intercepts, you set the quadratic function equal to zero and solve for x. For the equation f(x) = x^{2} - 9, the x-intercepts are found by setting f(x) to zero, which gives us the solutions x=3 and x=-3. These solutions mean our graph crosses the x-axis at the points (-3, 0) and (3, 0). Plotting the x-intercepts gives us specific points that help to shape the parabola on the graph.

Y-Intercept

The y-intercept refers to the point where the parabola crosses the y-axis. To find this, we plug x=0 into our quadratic function and solve for f(x). In our example, substituting x with zero in f(x) = x^{2} - 9 gives us f(0) = -9, which means the y-intercept is at the coordinate (0, -9). Coincidentally, the y-intercept is also the vertex in this particular example, a distinctive point where the graph changes direction. The y-intercept is particularly useful as it provides a starting point for graphing the quadratic function on a coordinate plane.