Vertex Form: Quadratic Function With Vertex (3, -7)

by Alex Johnson 52 views

In this article, we'll walk through how to determine the quadratic function in vertex form given the vertex of the parabola and another point it passes through. This is a common problem in algebra, and understanding the process can help you solve various quadratic function-related questions. Let's dive in!

Understanding Vertex Form

The vertex form of a quadratic function is expressed as:

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

Where:

  • (h, k) represents the vertex of the parabola.
  • a determines the direction the parabola opens (upwards if a > 0, downwards if a < 0) and the stretch or compression of the parabola.

Knowing the vertex form is crucial because it directly gives us the vertex coordinates, making it easier to analyze and graph quadratic functions. The vertex, as the name suggests, is the highest or lowest point on the parabola, which is a key feature.

Key Components of Vertex Form

To fully grasp vertex form, let's break down each component:

  • (h, k) - The Vertex: The vertex is the turning point of the parabola. If a > 0, the parabola opens upwards, and the vertex is the minimum point. Conversely, if a < 0, the parabola opens downwards, and the vertex is the maximum point. The coordinates (h, k) directly plug into the vertex form equation, giving us a starting point for graphing or analyzing the function.
  • a - The Stretch Factor: The coefficient a plays a pivotal role in determining the shape of the parabola. If |a| > 1, the parabola is stretched vertically, making it appear narrower. If 0 < |a| < 1, the parabola is compressed vertically, making it appear wider. The sign of a also tells us about the direction of the parabola: positive for upwards-opening and negative for downwards-opening. Understanding a helps in visualizing how the parabola will look on a graph.
  • (x - h) - Horizontal Shift: The (x - h) term inside the parenthesis indicates a horizontal shift of the parabola. If h is positive, the parabola shifts to the right by h units. If h is negative, the parabola shifts to the left by |h| units. This horizontal shift is crucial for positioning the parabola correctly on the coordinate plane.
  • + k - Vertical Shift: The + k term outside the parenthesis represents a vertical shift of the parabola. If k is positive, the parabola shifts upwards by k units. If k is negative, the parabola shifts downwards by |k| units. The vertical shift helps in determining the vertical position of the parabola's vertex.

Importance of Vertex Form

Why is vertex form so important? Here are a few reasons:

  1. Easy Identification of the Vertex: As mentioned earlier, the vertex (h, k) is immediately apparent from the equation.
  2. Graphing Made Simple: Vertex form makes graphing parabolas straightforward. Plot the vertex, use the stretch factor a to determine the shape, and you're good to go.
  3. Solving Optimization Problems: In many real-world scenarios, you might want to find the maximum or minimum value of a quadratic function. The vertex gives you this information directly.
  4. Converting from Standard Form: While the standard form y = ax^2 + bx + c is another way to represent quadratic functions, converting it to vertex form can provide additional insights, especially for identifying the vertex and understanding transformations.

By understanding the components and significance of vertex form, you can tackle a variety of quadratic function problems with confidence. Now, let’s apply this knowledge to solve the problem at hand.

Problem Setup

We are given:

  • Vertex: (3, -7)
  • Point on the graph: (1, -10)

We need to find the quadratic function in vertex form.

Steps to Solve

  1. Write the General Vertex Form:

    Start with the general form of a quadratic function in vertex form:

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

  2. Substitute the Vertex:

    Plug in the vertex (h, k) = (3, -7) into the equation:

    y = a(x - 3)^2 - 7

    Now we have a more specific form, but we still need to find the value of a.

  3. Use the Given Point:

    We know the graph passes through the point (1, -10). Substitute x = 1 and y = -10 into the equation:

    -10 = a(1 - 3)^2 - 7

  4. Solve for a:

    Simplify and solve the equation for a:

    -10 = a(-2)^2 - 7
-10 = 4a - 7
-3 = 4a
a = -3/4

    We've found that a = -3/4. This means the parabola opens downwards, and it's vertically compressed.

  5. Write the Final Equation:

    Substitute the value of a back into the vertex form equation:

    y = -3/4(x - 3)^2 - 7

    This is the quadratic function in vertex form that represents the graph with the given vertex and point.

Detailed Explanation of Each Step

  1. Writing the General Vertex Form: The vertex form equation, y = a(x - h)^2 + k, is the foundation of our solution. It's essential to start with this form because it directly incorporates the vertex coordinates, which are a key piece of information in this problem. This form allows us to build the quadratic function step by step using the given data. Remembering this general form is the first step in tackling any problem involving vertex form.

  2. Substituting the Vertex: By substituting the vertex coordinates (h, k) = (3, -7) into the general vertex form, we are effectively anchoring the parabola at its turning point. This substitution narrows down the possibilities, giving us y = a(x - 3)^2 - 7. Now, the only unknown is a, which determines the parabola's shape and direction. This step simplifies the problem significantly, allowing us to focus on finding just one variable.

  3. Using the Given Point: The point (1, -10) provides crucial additional information. By substituting x = 1 and y = -10 into the equation, we create a specific instance of the function that must hold true. This step transforms the problem into a simple algebraic equation: -10 = a(1 - 3)^2 - 7. We’re using the fact that any point on the graph of the function must satisfy the function’s equation.

  4. Solving for a: Solving for a involves simplifying and rearranging the equation. The steps include:

    • Simplifying (1 - 3)^2 to 4.
    • Adding 7 to both sides to isolate the term with a.
    • Dividing by 4 to find a = -3/4. This value of a tells us that the parabola opens downwards (a is negative) and is vertically compressed (the absolute value of a is less than 1). This is a critical step as it completes the determination of the function’s parameters.

  5. Writing the Final Equation: Substituting a = -3/4 back into the equation y = a(x - 3)^2 - 7 gives us the final quadratic function in vertex form: y = -3/4(x - 3)^2 - 7. This equation fully describes the parabola with the given vertex and passing through the specified point. We now have a complete representation of the quadratic function that satisfies all the given conditions.

Conclusion

The quadratic function in vertex form that represents the graph with a vertex at (3, -7) and passing through the point (1, -10) is:

y = -3/4(x - 3)^2 - 7

Understanding vertex form and how to use given information to find the specific equation is a valuable skill in algebra. Remember, the vertex form makes it easy to identify the vertex and, when combined with another point, allows you to determine the stretch factor a. This process can be applied to a variety of quadratic function problems.

For further exploration of quadratic functions and their properties, you might find resources at Khan Academy's Algebra Section to be very helpful.