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To convert a quadratic equation from vertex form ( y = a(x - h)^2 + k ) to standard form ( y = ax^2 + bx + c ), you need to expand the squared term. First, expand ( (x - h)^2 ) to get ( x^2 - 2hx + h^2 ). Then, multiply this by ( a ) and add ( k ), resulting in ( y = ax^2 - 2ahx + (ah^2 + k) ), which gives you the coefficients for ( ax^2 + bx + c ).
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Convert between standard and vertex form?
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How is the vertex form of an equation different from the standard form and what values change the shape of the graph?
The vertex form of a quadratic equation is expressed as ( y = a(x-h)^2 + k ), where ((h, k)) is the vertex of the parabola, while the standard form is ( y = ax^2 + bx + c ). In vertex form, the values of (a), (h), and (k) directly influence the shape and position of the graph; specifically, (a) determines the width and direction of the parabola, while (h) shifts it horizontally and (k) shifts it vertically. Changes to (a) affect the steepness, while altering (h) and (k) moves the vertex without changing the graph's shape.
What is the standard form of the equation of the parabola with vertex 00 and directrix y4?
Assuming the vertex is 0,0 and the directrix is y=4 x^2=0
How do you find the vertex of an equation in standard form?
To find the vertex of a quadratic equation in standard form, (y = ax^2 + bx + c), you can use the vertex formula. The x-coordinate of the vertex is given by (x = -\frac{b}{2a}). Once you have the x-coordinate, substitute it back into the equation to find the corresponding y-coordinate. The vertex is then the point ((-\frac{b}{2a}, f(-\frac{b}{2a}))).
How do you find the vertex of a parabola when the equation is in standard form?
To find the vertex of a parabola in standard form, which is given by the equation ( y = ax^2 + bx + c ), you can use the formula for the x-coordinate of the vertex: ( x = -\frac{b}{2a} ). Once you have the x-coordinate, substitute it back into the original equation to find the corresponding y-coordinate. The vertex will then be at the point ( (x, y) ).
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.
Convert between standard and vertex form?
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How is the vertex form of an equation different from the standard form and what values change the shape of the graph?
The vertex form of a quadratic equation is expressed as ( y = a(x-h)^2 + k ), where ((h, k)) is the vertex of the parabola, while the standard form is ( y = ax^2 + bx + c ). In vertex form, the values of (a), (h), and (k) directly influence the shape and position of the graph; specifically, (a) determines the width and direction of the parabola, while (h) shifts it horizontally and (k) shifts it vertically. Changes to (a) affect the steepness, while altering (h) and (k) moves the vertex without changing the graph's shape.
What are the advantages and disadvantages of each form of a formula of completing the square?
Completing the square can be expressed in two forms: the vertex form (y = a(x - h)^2 + k) and the standard form (y = ax^2 + bx + c). The vertex form highlights the vertex of the parabola, making it easy to graph and identify transformations, while the standard form is useful for identifying the coefficients and analyzing the general shape. However, the vertex form can be less intuitive for solving equations, whereas the standard form may require more steps for graphing or identifying the vertex. Each form serves different purposes depending on the problem at hand.
How do you convert standard form to vertex form?
y= -5/49(x-9)^2+5
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.
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 standard form of the equation of the parabola with vertex 00 and directrix y4?
Assuming the vertex is 0,0 and the directrix is y=4 x^2=0
How do you find the vertex of an equation in standard form?
To find the vertex of a quadratic equation in standard form, (y = ax^2 + bx + c), you can use the vertex formula. The x-coordinate of the vertex is given by (x = -\frac{b}{2a}). Once you have the x-coordinate, substitute it back into the equation to find the corresponding y-coordinate. The vertex is then the point ((-\frac{b}{2a}, f(-\frac{b}{2a}))).
How do you find the vertex of a parabola when the equation is in standard form?
To find the vertex of a parabola in standard form, which is given by the equation ( y = ax^2 + bx + c ), you can use the formula for the x-coordinate of the vertex: ( x = -\frac{b}{2a} ). Once you have the x-coordinate, substitute it back into the original equation to find the corresponding y-coordinate. The vertex will then be at the point ( (x, y) ).
When is it better to have the quadratic function in vertex form instead of standard form?
The quadratic function is better represented in vertex form when you need to identify the vertex of the parabola quickly, as it directly reveals the coordinates of the vertex ((h, k)). This form is particularly useful for graphing, as it allows you to see the maximum or minimum point of the function immediately. Additionally, if you're interested in transformations such as shifts and reflections, vertex form clearly outlines how the graph is altered.
How can you change 2.56 into standard form?
You have written it in standard form.
