Graphing Y=(x-3)^2-16: A Step-by-Step Guide

Dec 4, 2025 by Alex Johnson 44 views

Understanding how to graph quadratic equations is a fundamental skill in mathematics. This guide will walk you through the process of sketching the graph of the equation $y=(x-3)^2-16$ and identifying the correct graph from a set of options. We'll cover the key features of a parabola, how to find the vertex, axis of symmetry, and intercepts, and then use this information to accurately sketch the graph. Whether you're a student tackling algebra or just brushing up on your math skills, this guide will provide you with a clear and comprehensive approach to graphing quadratic functions.

Understanding the Quadratic Equation

The equation $y=(x-3)^2-16$ is a quadratic equation in vertex form. Understanding this form is crucial for sketching the graph efficiently. The general vertex form of a quadratic equation is given by:

$y = a(x-h)^2 + k$

Where:

$(h, k)$ represents the vertex of the parabola.
$a$ determines the direction and stretch of the parabola. If $a > 0$ , the parabola opens upwards, and if $a < 0$ , it opens downwards. The absolute value of $a$ dictates the vertical stretch; a larger $|a|$ means a narrower parabola, while a smaller $|a|$ means a wider parabola.

In our equation, $y=(x-3)^2-16$ , we can identify the values as follows:

$a = 1$ (since there is no coefficient explicitly written before the parentheses, it's understood to be 1).
$h = 3$
$k = -16$

Knowing these values allows us to immediately determine some key features of the graph. The vertex of the parabola is $(h, k) = (3, -16)$ . Since $a = 1$ is positive, the parabola opens upwards. This foundational understanding is the first step in accurately sketching the graph. We will now explore how to find other critical points such as the axis of symmetry and intercepts to further refine our sketch.

Finding the Vertex and Axis of Symmetry

As established earlier, the vertex of the parabola represented by the equation $y=(x-3)^2-16$ is a crucial point for sketching the graph. From the vertex form $y = a(x-h)^2 + k$ , we identified the vertex as $(h, k) = (3, -16)$ . The vertex is the point where the parabola changes direction – it's the minimum point if the parabola opens upwards (as in our case) and the maximum point if it opens downwards. Understanding the vertex gives us a central point around which to draw the rest of the parabola.

Additionally, every parabola has an axis of symmetry, which is a vertical line that passes through the vertex. This line divides the parabola into two symmetrical halves. The equation of the axis of symmetry is given by $x = h$ . For our equation, since $h = 3$ , the axis of symmetry is the vertical line $x = 3$ . This means that if we were to fold the graph along the line $x = 3$ , the two halves of the parabola would perfectly overlap. The axis of symmetry not only helps in visualizing the symmetry of the parabola but also aids in plotting points efficiently, as we know that for every point on one side of the axis, there's a corresponding point on the other side at the same vertical distance from the vertex.

Knowing the vertex $(3, -16)$ and the axis of symmetry $x = 3$ gives us a strong foundation for sketching the parabola. We now have a central point and a line of reflection to guide our drawing. The next step is to find the intercepts, which will give us additional points to make our sketch even more accurate. We will look at both the y-intercept and the x-intercepts in the following sections.

Determining the Intercepts

To accurately sketch the graph of $y=(x-3)^2-16$ , it's essential to find the intercepts: the points where the parabola intersects the x-axis and the y-axis. These points provide valuable information about the parabola's position and shape.

Finding the y-intercept

The y-intercept is the point where the graph intersects the y-axis. To find it, we set $x = 0$ in the equation and solve for $y$ . So, for our equation:

$y = (0-3)^2 - 16$ $y = (-3)^2 - 16$ $y = 9 - 16$ $y = -7$

Therefore, the y-intercept is the point $(0, -7)$ . This tells us that the parabola crosses the y-axis at the point where $y = -7$ .

Finding the x-intercepts

The x-intercepts are the points where the graph intersects the x-axis. To find them, we set $y = 0$ in the equation and solve for $x$ :

$0 = (x-3)^2 - 16$

First, we isolate the squared term:

$(x-3)^2 = 16$

Next, we take the square root of both sides, remembering to consider both positive and negative roots:

$x-3 = \\\pm\\\sqrt{16}$ $x-3 = \\\pm\\\ 4$

Now, we solve for $x$ in both cases:

Case 1: $x-3 = 4$ $x = 4 + 3$ $x = 7$

Case 2: $x-3 = -4$ $x = -4 + 3$ $x = -1$

So, the x-intercepts are $x = 7$ and $x = -1$ . This gives us the points $(7, 0)$ and $(-1, 0)$ , which are where the parabola crosses the x-axis.

By finding both the y-intercept and the x-intercepts, we've identified three key points that will help us accurately sketch the graph. We have the vertex $(3, -16)$ , the y-intercept $(0, -7)$ , and the x-intercepts $(7, 0)$ and $(-1, 0)$ . With these points and the understanding of the parabola's symmetry, we can proceed to sketching the graph.

Sketching the Graph

Now that we have identified the key features of the parabola $y=(x-3)^2-16$ , we can proceed to sketching the graph. We know the following:

Vertex: $(3, -16)$
Axis of Symmetry: $x = 3$
y-intercept: $(0, -7)$
x-intercepts: $(7, 0)$ and $(-1, 0)$

Follow these steps to sketch the graph:

Plot the Vertex: Start by plotting the vertex $(3, -16)$ on the coordinate plane. This is the lowest point on the parabola since it opens upwards.
Draw the Axis of Symmetry: Draw a vertical dashed line through $x = 3$ . This line helps maintain the symmetry of the parabola.
Plot the Intercepts: Plot the y-intercept $(0, -7)$ and the x-intercepts $(7, 0)$ and $(-1, 0)$ . These points provide additional guidance for the shape of the parabola.
Use Symmetry to Plot Additional Points: Since the parabola is symmetric about the axis of symmetry, for every point on one side, there's a corresponding point on the other side. For example, the y-intercept $(0, -7)$ is 3 units to the left of the axis of symmetry ( $x = 3$ ). Therefore, there should be a corresponding point 3 units to the right of the axis of symmetry, which is the point $(6, -7)$ .
Sketch the Parabola: Now, carefully sketch the parabola by drawing a smooth curve through the plotted points. The parabola should be symmetrical about the axis of symmetry and have a U-shape that opens upwards, as $a = 1$ is positive.
Verify the Shape: Double-check that your sketch aligns with the properties we identified earlier. The vertex should be the minimum point, the parabola should open upwards, and it should pass through all the plotted intercepts.

By following these steps, you can create an accurate sketch of the graph of $y=(x-3)^2-16$ . This visual representation provides a clear understanding of the function's behavior and its relationship to the coordinate plane. The next step is to identify which of the given options (Graph A, Graph B, Graph C, or Graph D) corresponds to your sketch.

Identifying the Correct Graph

After sketching the graph of $y=(x-3)^2-16$ , the final step is to identify the correct graph from the given options (A, B, C, and D). To do this, compare your sketch with each of the provided graphs, looking for key similarities and differences.

Consider the following features when comparing graphs:

Vertex: The vertex is a critical point. Ensure the graph you choose has a vertex at $(3, -16)$ .
Direction of Opening: The parabola opens upwards in our case. Eliminate any graphs that show a parabola opening downwards.
Intercepts: Check if the graph intersects the y-axis at $(0, -7)$ and the x-axis at $(-1, 0)$ and $(7, 0)$ .
Symmetry: The graph should be symmetrical about the line $x = 3$ . Visually inspect if the two halves of the parabola appear to be mirror images of each other across this line.
Overall Shape: Compare the general shape and width of your sketched parabola with the given options. Remember that the coefficient $a = 1$ affects the width; in this case, it's a standard parabola without any significant stretching or compression.

By systematically comparing these features, you can eliminate graphs that don't match and confidently identify the graph that corresponds to your sketch. This exercise not only reinforces your understanding of graphing quadratic equations but also develops your analytical skills in visual pattern matching.

In conclusion, accurately sketching and identifying the graph of $y=(x-3)^2-16$ involves understanding the vertex form of a quadratic equation, finding key features such as the vertex and intercepts, and then carefully plotting these points to create the graph. This comprehensive approach ensures that you can confidently tackle similar graphing problems in the future. For further learning on graphing quadratic functions and related concepts, you can visit Khan Academy's Algebra 1 curriculum.