Respuesta :
The quadratic equation are equations that have an axis of symmetry
passing through the vertex.
- 2. The quadratic equations are; y = x² + 2·x - 3 and y = -x² - 2·x + 3
- 3. The equation is; y = -x² - 2·x + 10
- 4. The equation is; y = x² + 3·x - 2
Reasons:
2. The x-intercepts are given by the point at which the y-value of the equation are zero.
Therefore;
The quadratic equation are;
(x + 3)·(x - 1) = 0
x² + 2·x - 3 = 0
The equation can also be written in the form;
(-x - 3)·(x - 1) = 0
-x² - 2·x + 3 = 0
The quadratic equations are;
- y = x² + 2·x - 3 and y = -x² - 2·x + 3
3. The points through which the quadratic equation passes are;
(-4, 2) and (1, 2)
The value at the vertex = Maximum value
Taking the vertex as the point midway between the two given points, we have;
[tex]\displaystyle Coordinates \ of \ the \ vertex = \left( \frac{-4 + 2}{2} , \, k \right) = \left(-1, \, k)[/tex]
The coordinates of the vertex, (h, k) = (-1, y)
The vertex form of a quadratic equation is presented as follows;
(y - k) = a·(x - h)²
y = a·(x - h)² + k
Which gives;
y = a·(x - (-1))² + k = a·(x + 1)² + k
y = a·(x + 1)² + k
At the point (-4, 2), we have;
2 = a·((-4) + 1)² + k = 9·a + k
2 = 9·a + k
Taking the value of k as 11, we have;
(h, k) = (-1, 11)
2 = 9·a + 11
[tex]\displaystyle a = \frac{2 - 11}{9} = -1[/tex]
Which gives;
y = -1·(x + 1)² + 11 = -x² - 2·x + 10
y = -x² - 2·x + 10
When x = -4, we have;
y = -(-4)² - 2·(-4) + 10 = 2
When x = 2, we have;
y = -(2)² - 2·(2) + 10 = 2
- The equation is; y = -x² - 2·x + 10
4. The points through which the graph passes are; (-4, 2) and (1, 2)
The x-coordinate of the minimum vertex is given by the equation;
[tex]\displaystyle Coordinates \ of \ the \ vertex, \ (h, \, k) = \left( \frac{-4 + 1}{2} , \, k\right) = \left(-1.5, \, k)[/tex]
(h, k) = (-1.5, y)
The vertex form of the equation of a quadratic equation is presented as follows;
y = a·(x - h)² + k
Which gives;
y = a·(x - (-1.5))² + k
y = a·(x + 1.5)² + k
[tex]\displaystyle a = \mathbf{\frac{y - k}{\left(x + 1.5\right)^2}}[/tex]
At the point (1, 2), we have;
[tex]\displaystyle a = \frac{2 - k}{\left(1 + 1.5\right)^2} = \frac{2 - k}{6.25}[/tex]
When k = -4.25, we have;
[tex]\displaystyle a = \frac{2 - k}{\left(1 + 1.5\right)^2} = \frac{2 - \left(-4.25 \right)}{6.25} = 1[/tex]
The equation is therefore;
y = 1·(x + 1.5)² - 4.25 = x² + 3·x - 2
y = x² + 3·x - 2
At the point where x = -4, we have;
y = (-4)² + 3·(-4) - 2 = 2
At the point where x = 1, we have;
y = (1)² + 3·(1) - 2 = 2
Therefore;
- The equation is; y = x² + 3·x - 2
Learn more here:
https://brainly.com/question/7160396
