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.
The quadratic function given is in vertex form 𝑦 = a(𝑥 - q)^2 + p, where (q, p) is the vertex of the parabola.
In this case, the vertex form is 𝑦 = 2(𝑥 - 1)^2 + 3, so a = 2 and q = 1.
To find the number of x-intercepts, we can look at the discriminant, which is the value under the square root in the quadratic formula.
The formula for the discriminant is Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic function in standard form (𝑦 = ax^2 + bx + c).
In this case, the quadratic function is already in vertex form, so we'll need to convert it to standard form to find the values of a, b, and c.
Expanding 𝑦 = 2(𝑥 - 1)^2 + 3, we get 𝑦 = 2(𝑥^2 -2𝑥 + 1) + 3.
Simplifying further, we have 𝑦 = 2𝑥^2 - 4𝑥 + 2 + 3.
Combining like terms, we get 𝑦 = 2𝑥^2 - 4𝑥 + 5.
Now we can see that a = 2, b = -4, and c = 5. Calculating the discriminant, we have Δ = (-4)^2 - 4(2)(5) = 16 - 40 = -24.
Since the discriminant is negative (Δ < 0), this means that the quadratic function has no x-intercepts.
clearly, there are no intercepts.
The vertex is at (1,3) and the parabola opens upward.
Apologies for the confusion in my previous response. You are correct.
The given quadratic function is in vertex form 𝑦 = 2(𝑥 - 1)^2 + 3, where a = 2 and q = 1.
The vertex of the parabola is (1, 3). Since the coefficient of the squared term (a = 2) is positive, the parabola opens upward.
Since the parabola does not intersect or touch the x-axis, there are no x-intercepts.
To find the number of x-intercepts for a quadratic function in vertex form, we need to analyze the value of the coefficient "a", which is the value in front of the squared term.
In this case, the quadratic function is given as 𝑦 = 2(𝑥 − 1)^2 + 3, where the coefficient "a" is 2.
The number of x-intercepts depends on the sign of "a":
1. If "a" is positive (greater than 0): The graph of the quadratic function opens upwards, and the parabola does not intersect the x-axis. Hence, the number of x-intercepts is zero.
2. If "a" is negative (less than 0): The graph of the quadratic function opens downwards, and the parabola intersects the x-axis at one point or two points. Hence, the number of x-intercepts is either one or two.
In our case, the value of "a" is 2, which is positive. Therefore, the number of x-intercepts for the given quadratic function is zero.