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 ).
2
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.
Assuming the vertex is 0,0 and the directrix is y=4 x^2=0
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}))).
Writing a quadratic equation in vertex form, ( y = a(x-h)^2 + k ), highlights the vertex of the parabola, making it easier to graph and identify key features like the maximum or minimum value. In contrast, standard form, ( y = ax^2 + bx + c ), is useful for quickly determining the y-intercept and applying the quadratic formula for finding roots. When working with vertex form, methods like completing the square can be employed to convert from standard form, while factoring or using the quadratic formula can be more straightforward when in standard form. Each form serves specific purposes depending on the analysis needed.
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.
2
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.
y= -5/49(x-9)^2+5
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.
You complete the squares. y = ax2 + bx + c = (ax2 + b/2a)2 + c - b2/(4a2)
Assuming the vertex is 0,0 and the directrix is y=4 x^2=0
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}))).
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.
You have written it in standard form.
look for the interceptions add these and divide it by 2 (that's the x vertex) for the yvertex you just have to fill in the x(vertex) however you can also use the formula -(b/2a)
Writing a quadratic equation in vertex form, ( y = a(x-h)^2 + k ), highlights the vertex of the parabola, making it easier to graph and identify key features like the maximum or minimum value. In contrast, standard form, ( y = ax^2 + bx + c ), is useful for quickly determining the y-intercept and applying the quadratic formula for finding roots. When working with vertex form, methods like completing the square can be employed to convert from standard form, while factoring or using the quadratic formula can be more straightforward when in standard form. Each form serves specific purposes depending on the analysis needed.