The problem in the image consists of two parts:

(a) Plot the graph of the quadratic function $y = x(10 - x)$. Mark the points of intersection with the axes and estimate the turning points.

(b) Calculate the points of intersection with the axes.

Let's break this down.

(a) Graph the quadratic function:

The given quadratic function is: $y = x(10 - x)$ which can be expanded as: $y = 10x - x^2$ This is a downward-opening parabola because the coefficient of $x^2$ is negative.

• Intersection with the axes:

  • x-intercepts occur where $y = 0$. $x(10 - x) = 0 \quad \Rightarrow \quad x = 0 \quad \text{or} \quad x = 10$ So, the parabola intersects the x-axis at points $(0, 0)$ and $(10, 0)$.

  • y-intercept occurs where $x = 0$. This gives: $y = 0(10 - 0) = 0$ So the y-intercept is also at $(0, 0)$.

• Turning point: The turning point of a parabola occurs at its vertex. For a quadratic of the form $y = ax^2 + bx + c$, the x-coordinate of the vertex is given by: $x_{\text{vertex}} = -\frac{b}{2a}$ Here, $a = -1$ and $b = 10$, so: $x_{\text{vertex}} = -\frac{10}{2(-1)} = 5$ Plugging $x = 5$ into the equation to find the y-coordinate: $y = 5(10 - 5) = 25$ So, the turning point is at $(5, 25)$.

(b) Points of intersection with the axes:

• x-intercepts: From part (a), the x-intercepts are at $x = 0$ and $x = 10$, so the points of intersection with the x-axis are $(0, 0)$ and $(10, 0)$.

• y-intercept: Since the equation $y = x(10 - x)$ passes through $(0, 0)$, the y-intercept is also $(0, 0)$.

Summary:

• The parabola has x-intercepts at $(0, 0)$ and $(10, 0)$.
• The vertex (turning point) is at $(5, 25)$.
• The y-intercept is $(0, 0)$.

Would you like more details, or have any questions?

Here are 5 related questions you could explore:

1. How do you determine whether a parabola opens upwards or downwards?
2. How can you find the axis of symmetry of a quadratic function?
3. What is the significance of the turning point in a quadratic graph?
4. How do we calculate the maximum or minimum value of a quadratic function?
5. Can you explain how completing the square would help in identifying the vertex?

Tip: When graphing a quadratic, it's often helpful to factor the equation if possible and find key points like intercepts and the vertex to guide your sketch.