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The standard form of quadratic function is: f(x) = a(x - h)^2 + k, a is different than 0
The graph of f is a parabola whose vertex it is the point (h, k). If a > 0, the parabola opens upward; if a < 0, the parabola opens downward. Furthermore, if |a| is small, the parabola opens more flatly than if |a| is large. It is a general procedure for graphing parabolas whose equations are in standard form:
Example 1:
Graph the the quadratic function f(x) = -2(x - 3)^2 + 8
Solution:
Standard form: f(x) = a(x - h)^2 + k Given function: f(x) = -2(x - 3) + 8
From the give function we have: a= -2; h= 3; k = 8
Step 1. Determine how the parabola opens.
Note that a = -2. Since a < 0, the parabola is open downward.
Step 2. Find the vertex.
The vertex of parabola is at (h, k). because h = 3 and k = 8, the parabola has its vertex at (3, 8).
Step 3. Find the x-intercepts by solving f(x) = 0. Replace f(x) with 0 at
f(x) = -2(x - 3)^2 + 8 and solve for x
0 = -2(x - 3)^2 + 8
2(x - 3)^2 = 8
(x- 3)^2 = 4
x - 3 = square radical 4
x - 3 = 2 or x -3 = -2
x = 5 or x = 1
The x- intercepts are 1 and 5. Thus the parabola passes through the points (1, 0) and (5, 0), this means that parabola intercepts the x-axis at 1 and 5.
Step 4. Find the y-intercept by computing f(0). Replace x with 0 in f(x) = _2(x - 3)^2 + 8
f(0) = -2(0 - 3)^2 + 8
f(0) = -2(9) + 8
f(0) = -10
The y-intercept is -10. Thus the parabola passes through the point (0, -10), this means that parabola intercepts the y-axis at -10.
Step 5. Graph the parabola. With a vertex at (3, 8), x-intercepts at 1 and 5, and a y-intercept at -10. The axis of symmetry is the vertical line whose equation is x = 3.
Example 2:
Graphing a quadratic function in the form f(x) = ax^2 + bx + c
Graph the quadratic function f(x) = -x^2 - 2x + 1
Solution:
Here a = -1, b = -2, and c = 1
Step 1. Determine how the parabola opens. Since a = 1, a < 0, the parabola opens downward.
Step 2. Find the vertex.
We know that x-coordinate of the vertex is x = -b/2a. Substitute a with -1 and b with -2 into the equation for the x-coordinate:
x = - b/2a
x= -(-2)/(2)(-1)
x = -1, so the x-coordinate of the vertex is -1, and the y-coordinate of the vertex will be f(-1). thus the vertex is at ( -1, f(-1) )
f(x) = -x^2 - 2x +1
f(-1) = -(-1)^2 - 2(-1) + 1
f(-1) = -1 + 2 + 1
f(-1) = 2
So the vertex of the parabola is (-1, 2)
Step 3. Find the x-intercepts by solving f(x) = o
f(x) = -x^2 -2x + 1
0 = -x^2- 2x + 1
We can't solve this equation by factoring, so we use the quadratic formula to solve it.
we get to solution: One solution is x = -2.4 and the other solution is 0.4 (approximately). Thus the x-intercepts are approximately -2.4 and 0.4. The parabola passes through ( -2.4, 0) and (0.4, 0)
Step 4. Find the y-intercept by computing f(0).
f(x) = -x^2 - 2x + 1
f(0) = -(0)^2 - 2(0) + 1
f(0) = 1
The y-intercept is 1. The parabola passes through (0, 1).
Step 5. graph the parabola with vertex at (-1, 2), x-intercepts approximately at -2.4 and 0.4, and y -intercept at 1. The line of symmetry is the vertical line with equation
x= -1.
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In the vertex form of a quadratic function y equals a the quantity of x-b squared plus c what does the b tell you about the graph?
The standard form of the quadratic function in (x - b)2 + c, has a vertex of (b, c). Thus, b is the units shifted to the right of the y-axis, and c is the units shifted above the x-axis.
How do you graph y equals x squared minus 4?
its a simple parobola symmetric about y axis, having its vertex at (0,-4). we can make its graph by changing its equation in standard form so that we can get its different standard points like vertex, focus, etc.
What is y equals x squared?
it will form a parabola on the graph with the vertex at point (0,0) and points at (1,1), (-1,1), (2,4), (-2,4)......
How do you change a quadratic formula from standard to vertex form?
You complete the squares. y = ax2 + bx + c = (ax2 + b/2a)2 + c - b2/(4a2)
What is the difference between standard form and vertex form?
The difference between standard form and vertex form is the standard form gives the coefficients(a,b,c) of the different powers of x. The vertex form gives the vertex 9hk) of the parabola as part of the equation.
What different information do you get from vertex form and quadratic equation in standard form?
The graph of a quadratic function is always a parabola. If you put the equation (or function) into vertex form, you can read off the coordinates of the vertex, and you know the shape and orientation (up/down) of the parabola.
What is the definition of a Vertex form of a quadratic function?
it is a vertices's form of a function known as Quadratic
What is the equation for vertex form?
The vertex form for a quadratic equation is y=a(x-h)^2+k.
How do you convert vertex form to quadratic form?
Do you have a specific vertex fraction? vertex = -b/2a wuadratic = ax^ + bx + c
How can you use intercept form?
You can use intercept form to graph and write quadratic functions. y=a(x-p)(x-q) You can also use intercept form to graph and write cubic functions. y=a(x-p)(x-q)(x-r)
In the vertex form of a quadratic function y equals a the quantity of x-b squared plus c what does the b tell you about the graph?
The standard form of the quadratic function in (x - b)2 + c, has a vertex of (b, c). Thus, b is the units shifted to the right of the y-axis, and c is the units shifted above the x-axis.
Why is it better to have the quadratic function in vertex form instead of standard form?
If you want to graph the function, it is quite easy: y=a(x-h)2-k . . . you can plot the vertex (h,k); the 'a' tells you how wide or narrow the u-shape is, and whether it opens up or down.
What part of speech is quadratic function?
A quadratic function is a noun. The plural form would be quadratic functions.
What is the formula for quadratic equation in vertex form?
y=2(x-3)+1
What determines whether the graph of the quadratic function will open upward or downward?
The slope of your quadratic equation in general form or standard form.
What part of speech is function?
A quadratic function is a noun. The plural form would be quadratic functions.
What is the difference between linear and quadratic equations?
A linear equation has the form of mx + b, while a quadratic equation's form is ax2+bx+c. Also, a linear equation's graph forms a line, while a quadratic equation's graph forms a parabola.
