Graph the quadratic function by plotting the vertex and some intercepts:

𝑓
(
𝑥
)
=
4
𝑥
2
−
8
𝑥
+
3

Question

Graph the quadratic function by plotting the vertex and some intercepts:

𝑓
(
𝑥
)
=
4
𝑥
2
−
8
𝑥
+
3

BlackTom AI · Accepted Answer

We are given the quadratic function f(x) = 4x^2 − 8x + 3 to graph by plotting the vertex and intercepts.

First, identify key features of the parabola:
- a = 4 (positive), so the parabola opens upward.
- b = −8, c = 3.

Vertex:
- The x-coordinate of the vertex is x = −b/(2a) = −(−8)/(2×4) = 8/8 = 1.
- The y-coordinate is f(1) = 4(1)^2 − 8(1) + 3 = 4 − 8 + 3 = −1.
- Therefore, the vertex is at (1, −1).

X-intercepts (roots): set f(x) = 0 and solve 4x^2 − 8x + 3 = 0.
- Compute the discriminant: Δ = (−8)^2 − 4·4·3 = 64 − 48 = 16.
- Roots: x = [−(−8) ± √Δ] / (2a) = [8 ± 4] / 8, giving x = 12/8 = 3/2 = 1.5 and x = 4/8 = 1/2 = 0.5.
- So the x-intercepts are at (0.5, 0) and (1.5, 0).

Y-intercept:
- Set x = 0: f(0) = 3. So the y-intercept is (0, 3).

Summary of key points to plot on the graph:
- Vertex at (1, −1).
- X-intercepts at (0.5, 0) and (1.5, 0).
- Y-intercept at (0, 3).

If the question presents multiple-choice options, you would compare these computed features (vertex, intercepts) to the options to determine which graph correctly reflects these values. In the provided options, there are no visible choices to evaluate, so the correct graph would be the one passing through the points (1, −1), (0.5, 0), (1.5, 0), and (0, 3) with the parabola opening upward.

MATH136-1IP-IO-202430-I-81X

Graph the quadratic function by plotting the vertex and some intercepts: 𝑓 ( 𝑥 ) = 4 𝑥 2 − 8 𝑥 + 3

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