1. The figure below represents the graph of the function $$ y=x^{2}+4 x-5 $$. (a) Use the given tangents to find the gradi- ents of the curve at (i) $$ A $$ (ii) $$ B $$ (b) Use the graph to find the roots of the function. (c) State the equation of the line of symme- try of the curve. (d) For what value of $$ x $$ is the gradient of the curve equal to zero?

Question

UpStudy AI · Accepted Answer

Let's analyze the function $$ y = x^{2} + 4x - 5 $$ and address each part of the problem step by step.

### Part (a): Finding the Gradients at Points $$ A $$ and $$ B $$

To determine the gradients (slopes) of the curve at specific points, we first need to find the derivative of the function.

**1. Derivative of the Function:**
$$
\frac{dy}{dx} = 2x + 4
$$
This derivative represents the gradient of the curve at any point $$ x $$.

**2. Assuming Points $$ A $$ and $$ B $$ are the Roots:**

Given that parts of the problem reference tangents and a graph (which isn't provided here), we'll make a reasonable assumption that points $$ A $$ and $$ B $$ correspond to the roots of the function.

**3. Finding the Roots:**
$$
x^{2} + 4x - 5 = 0 \
x = \frac{-4 \pm \sqrt{16 + 20}}{2} = \frac{-4 \pm 6}{2}
$$
$$
\Rightarrow x = 1 \quad \text{and} \quad x = -5
$$
So, let's assume:
- Point $$ A $$ has $$ x = 1 $$
- Point $$ B $$ has $$ x = -5 $$

**4. Calculating the Gradients:**
$$
\text{Gradient at } A: \frac{dy}{dx}\bigg|_{x=1} = 2(1) + 4 = 6
$$
$$
\text{Gradient at } B: \frac{dy}{dx}\bigg|_{x=-5} = 2(-5) + 4 = -6
$$

### Summary for Part (a):
- **(i) Gradient at $$ A $$:** $$ 6 $$
- **(ii) Gradient at $$ B $$:** $$ -6 $$

### Part (b): Finding the Roots of the Function

As derived earlier, the roots of the function $$ y = x^{2} + 4x - 5 $$ are:
$$
x = 1 \quad \text{and} \quad x = -5
$$

### Part (c): Equation of the Line of Symmetry

For a quadratic function $$ y = ax^{2} + bx + c $$, the axis of symmetry is given by:
$$
x = -\frac{b}{2a}
$$
Here, $$ a = 1 $$ and $$ b = 4 $$:
$$
x = -\frac{4}{2 \times 1} = -2
$$
**Equation of the Line of Symmetry:** $$ x = -2 $$

### Part (d): Value of $$ x $$ Where Gradient is Zero

To find the $$ x $$-value where the gradient is zero, set the derivative equal to zero:
$$
2x + 4 = 0 \
2x = -4 \
x = -2
$$
This confirms that the minimum point of the parabola (where the curve changes direction) occurs at $$ x = -2 $$, which aligns with the axis of symmetry.

---

### **Final Answers:**

**(a)**
- **(i) Gradient at $$ A $$:** 6
- **(ii) Gradient at $$ B $$:** –6

**(b)** The roots of the function are $$ x = 1 $$ and $$ x = -5 $$.

**(c)** The line of symmetry is the vertical line $$ x = -2 $$.

**(d)** The gradient of the curve is zero when $$ x = -2 $$.
**(a)** - **(i) Gradient at $$ A $$:** 6 - **(ii) Gradient at $$ B $$:** –6 **(b)** The roots are $$ x = 1 $$ and $$ x = -5 $$. **(c)** The line of symmetry is $$ x = -2 $$. **(d)** The gradient is zero at $$ x = -2 $$.

Upstudy AI Solution

Answer

Solution

Part (a): Finding the Gradients at Points and

Summary for Part (a):

Part (b): Finding the Roots of the Function

Part ©: Equation of the Line of Symmetry

Part (d): Value of Where Gradient is Zero

Final Answers:

Bonus Knowledge

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