Graph y=10-x^2+3x
The given problem asks you to plot the graph of a quadratic equation, y = 10 - x^2 + 3x, on a coordinate plane. This entails identifying the shape of the graph, which will be a parabola, and determining specific features such as the vertex, axis of symmetry, and intercepts, in order to accurately sketch the curve that represents all the solutions to the equation in the Cartesian coordinate system.
Rearrange the equation to standard form:
Analyze the parabola's characteristics.
Convert the quadratic equation to vertex form.
Complete the square for the quadratic term
Identify
Recall the vertex form of a parabola:
Calculate
Plug in the values for
Simplify to find
Determine
Insert values for
Simplify to find
Combine like terms:
Insert
Identify
The parabola opens downwards as
The vertex is at
Calculate the focal distance
Find the focus by adjusting the vertex's
The axis of symmetry is the vertical line through the vertex:
Determine the directrix, a horizontal line, using
Summarize the parabola's properties for graphing.
Select points around the vertex and calculate their corresponding
Calculate
Plot the parabola using the vertex, focus, axis of symmetry, directrix, and calculated points.
Draw the parabola on a graph with the identified properties and points.
Standard Form of a Quadratic Equation: The standard form is
Vertex Form of a Quadratic Equation: The vertex form is
Completing the Square: A method used to convert a quadratic equation into vertex form by adding and subtracting a particular value to create a perfect square trinomial.
Parabola Properties:
The vertex is the highest or lowest point on the parabola, depending on whether it opens up or down.
The axis of symmetry is a vertical line that passes through the vertex.
The focus is a point inside the parabola where all the reflected rays (if the parabola were a mirror) would meet.
The directrix is a line perpendicular to the axis of symmetry that is the same distance from the vertex as the focus but outside the parabola.
Graphing a Parabola: To graph a parabola, you need the vertex, the direction it opens (up or down), the axis of symmetry, and a few points on the parabola. The focus and directrix can also help in sketching a more accurate graph.