A quadratic function is given in vertex form: 𝑦 = 2(𝑥 − 1)^2 + 3.
a) Use the values of a and q to find the number of x-intercepts.
b) Change the function to standard form and use the discriminant to check your conclusion from part (a) of this question.
In vertex form, a quadratic function is given by
y = a(x - q)^2 + p
where (q, p) is the vertex of the parabola.
a) Since the function is in vertex form, we can see that a = 2.
To find the number of x-intercepts, we need to determine if the parabola intersects the x-axis or not.
If the parabola intersects the x-axis, it will have two x-intercepts.
If the parabola does not intersect the x-axis, it will have zero x-intercepts.
To determine this, we can check the value of a.
If a is positive (a > 0), the parabola opens upward and does not intersect the x-axis, so it will have zero x-intercepts.
If a is negative (a < 0), the parabola opens downward and intersects the x-axis in two points, so it will have two x-intercepts.
In this case, a = 2, which is positive, so the parabola does not intersect the x-axis. Therefore, it has zero x-intercepts.
b) To change the function from vertex form to standard form, we need to expand and simplify.
y = 2(x - 1)^2 + 3
Using the formula (a - b)^2 = a^2 - 2ab + b^2, we can expand the expression.
y = 2(x^2 - 2x + 1) + 3
y = 2x^2 - 4x + 2 + 3
y = 2x^2 - 4x + 5
Now the function is in standard form.
The standard form of a quadratic function is given by:
y = ax^2 + bx + c
where a, b, and c are constants.
Comparing the standard form obtained with the general form, we can see that:
a = 2
b = -4
c = 5
Now, we can use the discriminant to determine the number of x-intercepts.
The discriminant of a quadratic equation is given by:
D = b^2 - 4ac
In this case, the discriminant is:
D = (-4)^2 - 4(2)(5)
D = 16 - 40
D = -24
If the discriminant is positive (D > 0), the quadratic equation has two distinct real roots and therefore, two x-intercepts.
If the discriminant is zero (D = 0), the quadratic equation has one real root and therefore, one x-intercept.
If the discriminant is negative (D < 0), the quadratic equation has no real roots and therefore, no x-intercepts.
In this case, the discriminant is negative (D = -24), so there are no x-intercepts.
This conclusion confirms our earlier answer from part (a) where we found that the parabola given by the vertex form does not intersect the x-axis and therefore, has zero x-intercepts.