ClassX Video Lessons Subjects Algebra Graphing a parabola in vertex form | Quadratic equations | Algebra I
In this article, we’re going to learn how to graph the quadratic equation given by y = -2(x – 2)2 + 5. We’ll find important features of the parabola, like its vertex and other points needed to draw the complete graph.
The equation y = -2(x – 2)2 + 5 is in what’s called vertex form. This form helps us easily find the vertex and see which way the parabola opens. The part (x – 2)2 is always zero or positive. But because it’s multiplied by -2, the whole expression will always be zero or negative.
The highest value of y happens when the squared part is zero. This occurs when x – 2 = 0, or x = 2. Plugging x = 2 back into the equation gives us:
y = -2(0)2 + 5 = 5
So, the highest point of the parabola is at the vertex (2, 5).
To draw the parabola, we start by plotting the vertex at the point (2, 5). On a graph, this point is located at:
To draw the parabola accurately, we need at least three points. We already have the vertex, so let’s find two more points that are the same distance from the vertex. We can choose x = 1 and x = 3 for this.
For x = 1:
y = -2(1 – 2)2 + 5 = -2(-1)2 + 5 = -2 + 5 = 3
This means the point (1, 3) is on the parabola.
For x = 3:
y = -2(3 – 2)2 + 5 = -2(1)2 + 5 = -2 + 5 = 3
This gives us the point (3, 3).
Now we have three important points to graph the parabola:
With the points (1, 3), (2, 5), and (3, 3) plotted, we can sketch the parabola. The graph opens downwards because of the negative sign in front of the squared term, and the vertex is the highest point of the parabola.
In conclusion, we’ve successfully graphed the quadratic equation y = -2(x – 2)2 + 5 by finding its vertex and other points, giving us a clear picture of its shape and features.
Start by identifying the vertex of the quadratic equation y = -2(x – 2)2 + 5. Plot this point on graph paper. Discuss with your classmates why this point is the highest point on the graph and how the vertex form of the equation makes it easy to find.
Using the vertex and the additional points (1, 3) and (3, 3), draw the parabola on graph paper. Make sure to label the vertex and other points clearly. Compare your graph with a partner’s to see if they match.
Use a graphing calculator or an online graphing tool to input the equation y = -2(x – 2)2 + 5. Experiment by changing the coefficients and observe how the graph changes. Discuss how each part of the equation affects the shape and position of the parabola.
Create a piece of art using parabolas. Use different quadratic equations to draw various parabolas on a large sheet of paper. Color them in to create an artistic representation of quadratic functions. Share your artwork with the class and explain the equations you used.
Research real-world objects or phenomena that have parabolic shapes, such as satellite dishes or bridges. Present your findings to the class, explaining how the quadratic equation helps model these shapes and why the parabola is an efficient design.
Graphing – The process of plotting points or drawing lines on a coordinate plane to represent mathematical equations or data. – Example sentence: In algebra class, we practiced graphing linear equations to better understand their slopes and intercepts.
Quadratic – A type of polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants and a is not zero. – Example sentence: The quadratic equation x² – 4x + 4 = 0 can be solved by factoring or using the quadratic formula.
Equation – A mathematical statement that asserts the equality of two expressions, typically involving variables and constants. – Example sentence: Solving the equation 3x + 5 = 20 helps us find the value of x that makes the statement true.
Vertex – The highest or lowest point on the graph of a parabola, where the direction changes. – Example sentence: The vertex of the parabola y = x² – 4x + 3 is at the point (2, -1).
Parabola – A symmetric curve formed by the graph of a quadratic function, typically shaped like an open bowl. – Example sentence: The trajectory of a thrown ball can be modeled by a parabola on a graph.
Points – Specific locations on a graph represented by coordinates (x, y) that indicate their position on the plane. – Example sentence: To draw the line, we plotted the points (1, 2) and (3, 4) on the graph.
Maximum – The highest point or value that a function reaches, often found at the vertex of a downward-opening parabola. – Example sentence: The maximum value of the function y = -x² + 4x + 1 occurs at the vertex.
Value – The numerical quantity represented by a variable or expression in a mathematical equation. – Example sentence: By substituting x = 3 into the equation, we can find the value of y.
Calculate – To determine a numerical result using mathematical operations and procedures. – Example sentence: We need to calculate the slope of the line passing through the points (2, 3) and (4, 7).
Axis – A reference line on a graph, typically the horizontal x-axis or the vertical y-axis, used to measure coordinates. – Example sentence: The x-axis and y-axis intersect at the origin, which is the point (0, 0) on the graph.
