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.
$y = 10 - x^{2} + 3 x$
Answer
Solution:
Step:1
Rearrange the equation to standard form: $y = -x^2 + 3x + 10$.
Step:2
Analyze the parabola's characteristics.
Step:2.1
Convert the quadratic equation to vertex form.
Step:2.1.1
Complete the square for the quadratic term $-x^2 + 3x + 10$.
Step:2.1.1.1
Identify $a$, $b$, and $c$ from the standard form $ax^2 + bx + c$:
$a = -1$, $b = 3$, $c = 10$.
Step:2.1.1.2
Recall the vertex form of a parabola: $a(x + d)^2 + e$.
Step:2.1.1.3
Calculate $d$ using $d = \frac{b}{2a}$.
Step:2.1.1.3.1
Plug in the values for $a$ and $b$: $d = \frac{3}{2 \cdot -1}$.
Step:2.1.1.3.2
Simplify to find $d$: $d = -\frac{3}{2}$.
Step:2.1.1.4
Determine $e$ using $e = c - \frac{b^2}{4a}$.
Step:2.1.1.4.1
Insert values for $c$, $b$, and $a$: $e = 10 - \frac{3^2}{4 \cdot -1}$.
Step:2.1.1.4.2
Simplify to find $e$:
$e = 10 + \frac{9}{4}$.
Step:2.1.1.4.2.5
Combine like terms: $e = \frac{40 + 9}{4}$.
Step:2.1.1.5
Insert $a$, $d$, and $e$ into the vertex form: $y = -\left(x - \frac{3}{2}\right)^2 + \frac{49}{4}$.
Step:2.2
Identify $a$, $h$, and $k$ from the vertex form: $a = -1$, $h = \frac{3}{2}$, $k = \frac{49}{4}$.
Step:2.3
The parabola opens downwards as $a$ is negative.
Step:2.4
The vertex is at $(h, k) = \left(\frac{3}{2}, \frac{49}{4}\right)$.
Step:2.5
Calculate the focal distance $p$ using $\frac{1}{4a}$.
Step:2.6
Find the focus by adjusting the vertex's $y$-coordinate by $p$: Focus at $\left(\frac{3}{2}, 12\right)$.
Step:2.7
The axis of symmetry is the vertical line through the vertex: $x = \frac{3}{2}$.
Step:2.8
Determine the directrix, a horizontal line, using $y = k - p$: Directrix at $y = \frac{25}{2}$.
Step:2.9
Summarize the parabola's properties for graphing.
Step:3
Select points around the vertex and calculate their corresponding $y$ values.
Step:3.1 - Step:3.12
Calculate $y$ for $x$ values of $1$, $0$, $3$, and $4$.
Step:3.13
Plot the parabola using the vertex, focus, axis of symmetry, directrix, and calculated points.
Step:4
Draw the parabola on a graph with the identified properties and points.
Knowledge Notes:
Standard Form of a Quadratic Equation: The standard form is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
Vertex Form of a Quadratic Equation: The vertex form is $a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
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.
