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### Write the equation of a parabola given a vertex and point

Write the equation of a parabola given a vertex and point

Write the equation of a parabola given a vertex and point

### Which equation has a graph that is a parabola with a vertex at (-1, -1)? ^{[1]}

Which equation has a graph that is a parabola with a vertex at (-1, -1)?. Therefore, the equation of parabola is y = (x + 1)2 – 1.

The equation of a parabola with a vertex at (-1, -1) is y = (x +1)2 – 1.

### Parabolas ^{[2]}

A quadratic function is a function that can be written in the form where , and are real numbers and . This form is called the standard form of a quadratic function.

The graph of the equation , shown below, is a parabola. (Note that this is a quadratic function in standard form with and .)

In this case the vertex is the minimum, or lowest point, of the parabola. A large positive value of makes a narrow parabola; a positive value of which is close to makes the parabola wide.

### Which Equation Has A Graph That Is A Parabola With A Vertex At (-1,-1)? Y = (x – 1)2 + 1 Y = (x – 1)2 ^{[3]}

What is the result of converting 1500 yards into miles?. Dave bought the book when it was on sale for 20% off

Assume that the lines that appear to be parallel, are parallel. the x mean that you have to multiply and if you multiply 67 x 7 = 469

You can multiply the fraction to get 40/200, then simplify it to get 20/100 or 20%.. Why is it importance of sum and product of roots of a quadratic equation?

### Graphing Quadratic Equations Using the Axis of Symmetry ^{[4]}

Graphing Quadratic Equations Using the Axis of Symmetry. A quadratic equation is a polynomial equation of degree

The axis of symmetry of this parabola will be the line . The axis of symmetry passes through the vertex, and therefore the -coordinate of the vertex is

Substitute few more -values in the equation to get the corresponding -values and plot the points. Compare the equation with to find the values of , , and .

### How to find the Equation of a Parabola ^{[5]}

We learn how to find the equation of a parabola by writing it in vertex form. In the previous section, we learnt how to write a parabola in its vertex form and saw that a parabola’s equation: \[y = ax^2+bx+c\] could be re-written in vertex form: \[y = a\begin{pmatrix}x – h \end{pmatrix}^2+k\] where:

The idea is to use the coordinates of its vertex (maximum point, or minimum point) to write its equation in the form \(y=a\begin{pmatrix}x-h\end{pmatrix}^2+k\) (assuming we can read the coordinates \(\begin{pmatrix}h,k\end{pmatrix}\) from the graph) and then to find the value of the coefficient \(a\).. This is explained in the step-by-stem mathod, below, as well as in the tutorials.

This two-step method is better/further explained in the following tutorial, take the time to watch it now.. In the following tutorial we learn how to find a parabola’s using the coordinates of its vertex as well as the coordinates of its \(y\)-intercept.

### Transformations of the parabola ^{[6]}

We can translate the parabola vertically to produce a new parabola that is similar to the basic parabola. The function \(y=x^2+b\) has a graph which simply looks like the standard parabola with the vertex shifted \(b\) units along the \(y\)-axis

Similarly, we can translate the parabola horizontally. The function \(y=(x-a)^2\) has a graph which looks like the standard parabola with the vertex shifted \(a\) units along the \(x\)-axis

These two transformations can be combined to produce a parabola which is congruent to the basic parabola, but with vertex at \((a,b)\).. For example, the parabola \(y=(x-3)^2+4\) has its vertex at \((3,4)\) and its axis of symmetry has the equation \(x=3\).

### Parabolas ^{[7]}

The graph of a quadratic equation in two variables (y = ax2 + bx + c ) is called a parabola. The following graphs are two typical parabolas their x-intercepts are marked by red dots, their y-intercepts are marked by a pink dot, and the vertex of each parabola is marked by a green dot:

In order to graph a parabola we need to find its intercepts, vertex, and which way it opens.. Given y = ax2 + bx + c , we have to go through the following steps to find the points and shape of any parabola:

### Quadratic Functions ^{[8]}

251 #1-8, 10, 11, 15, 16, 18, 19, 21, 23, 24, 30, 33, 37, 38, 75. A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero.

Parabolas may open upward or downward and vary in “width” or “steepness”, but they all have the same basic “U” shape. The picture below shows three graphs, and they are all parabolas.

A parabola intersects its axis of symmetry at a point called the vertex of the parabola.. This means that if you are given any two points in the plane, then there is one and only one line that contains both points

### Characteristics of Parabolas ^{[9]}

– Identify the vertex, axis of symmetry, [latex]y[/latex]-intercept, and minimum or maximum value of a parabola from it’s graph.. – Identify a quadratic function written in general and vertex form.

– Define the domain and range of a quadratic function by identifying the vertex as a maximum or minimum.. The graph of a quadratic function is a U-shaped curve called a parabola

If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value

### Illustrative Mathematics ^{[10]}

– Find all quadratic functions described by the equation $y = ax^2 + bx + c$ whose graph contains the two points $(1,0)$ and $(3,0)$. How are the graphs of these functions related to one another?

How are the graphs of these functions related to one another?. The equation $$ f(x) = ax^2 + bx + c $$ has three constants $a,b,c$ which can take any real number value

Requiring the graph of $f$ to pass through a point $P_1$ puts one condition on $a,b,c$ while requiring the graph of $f$ to pass through $P_2$ puts a second condition on $a,b,c$. Assuming the $P_1$ and $P_2$ have distinct $x$-coordinates, this leaves one degree of freedom for the coefficients $a,b,c$ of a quadratic whose graph goes through $P_1$ and $P_2$

### Vertex Formula with Solved Examples ^{[11]}

In Mathematics, the vertex formula helps to find the vertex coordinate of a parabola, when the graph crosses its axes of symmetry. Generally, the vertex point is represented by (h, k)

Here, if the coefficient of x2 is positive, the vertex should be at the bottom of the U-shaped curve. If the coefficient of x2 is negative, then the vertex should be at the top of the U-shaped curve

We know that the standard form of the parabola is y=ax2+bx+c.. Thus, the vertex form of a parabola is y = a(x-h)2 + k.

### Sources

- https://www.cuemath.com/questions/which-equation-has-a-graph-that-is-a-parabola-with-a-vertex-at-1-1/
- https://www.varsitytutors.com/hotmath/hotmath_help/topics/parabolas
- https://oktrails.rcs.ou.edu/answers/4853600-which-equation-has-a-graph-that-is
- https://www.varsitytutors.com/hotmath/hotmath_help/topics/graphing-quadratic-equations-using-the-axis-of-symmetry
- https://www.radfordmathematics.com/functions/quadratic-functions-parabola/vertex-form/vertex-form-finding-equation-parabola.html
- https://amsi.org.au/ESA_Senior_Years/SeniorTopic2/2a/2a_2content_2.html
- http://www.csun.edu/~ayk38384/notes/mod11/Parabolas.html
- http://dl.uncw.edu/digilib/Mathematics/Algebra/mat111hb/PandR/quadratic/quadratic.html
- https://courses.lumenlearning.com/waymakercollegealgebra/chapter/characteristics-of-parabolas/
- https://tasks.illustrativemathematics.org/content-standards/tasks/379
- https://byjus.com/vertex-formula/