The given function is $y = 3x^2 + 5x + 1$, which is a quadratic function of the form $y = ax^2 + bx + c$, where $a = 3$, $b = 5$, and $c = 1$. Let’s analyze its characteristics:

1. Direction of the Parabola (Curve Opening)

Since $a = 3$, and it is positive, the parabola opens upward.

2. Vertex of the Parabola

The x-coordinate of the vertex is given by the formula: $x_{\text{vertex}} = \frac{-b}{2a} = \frac{-5}{2(3)} = \frac{-5}{6}.$ Since $x_{\text{vertex}} = -\frac{5}{6}$, the vertex is on the left side of the y-axis.

3. Y-intercept

The y-intercept is the value of $y$ when $x = 0$, which is: $y = 3(0)^2 + 5(0) + 1 = 1.$ Thus, the graph crosses the y-axis at $y = 1$.

4. X-intercepts

To find the x-intercepts (where the graph crosses the x-axis), solve the quadratic equation: $3x^2 + 5x + 1 = 0.$ Using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-5 \pm \sqrt{5^2 - 4(3)(1)}}{2(3)} = \frac{-5 \pm \sqrt{25 - 12}}{6} = \frac{-5 \pm \sqrt{13}}{6}.$ Since the discriminant ($\sqrt{13}$) is positive, there are two distinct real roots, meaning the graph intersects the x-axis at two distinct points.

Summary of the Characteristics:

• The vertex is to the left of the y-axis (i.e., "Titik puncak di kiri sumbu Y").
• The parabola opens upward.
• The graph crosses the x-axis at two distinct points.
• The graph crosses the y-axis above the x-axis (i.e., at $y = 1$).

Correct Answers:

• Titik puncak di kanan sumbu Y. (False, it's on the left.)
• Kurva terbuka menghadap ke atas. (True, it opens upward.)
• Memotong sumbu Y di bawah sumbu X. (False, it intersects at $y = 1$, which is above the x-axis.)
• Memotong sumbu X di dua titik berbeda. (True, it has two distinct x-intercepts.)

Would you like further details or explanations?

Related Questions:

1. How do you determine if a parabola opens upward or downward?
2. How do you find the vertex of a quadratic function?
3. What does the discriminant of a quadratic equation indicate?
4. How can you tell if a parabola will intersect the x-axis at two points, one point, or not at all?
5. What is the significance of the y-intercept in a quadratic graph?

Tip:

The discriminant ($b^2 - 4ac$) is key to determining the number of x-intercepts in a quadratic function.