3. Graph each quadratic relation and determine i) the equation of the axis of symmetry ii) the coordinates of the vertex \( \begin{array}{ll}\text { iii) the } y \text {-intercept } \\ \text { iv) the zeros } \\ \text { a) } y=x^{2}-8 x & \text { b) } y=x^{2}+2 x-15\end{array} \)

Function by following steps: - step0: Find the y-intercept: \(y=x^{2}-8x\) - step1: Set \(x\)=0\(:\) \(y=0^{2}-8\times 0\) - step2: Multiply: \(y=0^{2}-0\) - step3: Simplify: \(y=0\) Analyze the y intercept of the function \( y=x^{2}+2x-15 \) Function by following steps: - step0: Find the y-intercept: \(y=x^{2}+2x-15\) - step1: Set \(x\)=0\(:\) \(y=0^{2}+2\times 0-15\) - step2: Multiply: \(y=0^{2}+0-15\) - step3: Simplify: \(y=-15\) Analyze the x intercept of the function \( y=x^{2}-8x \) Function by following steps: - step0: Find the \(x\)-intercept/zero: \(y=x^{2}-8x\) - step1: Set \(y\)=0\(:\) \(0=x^{2}-8x\) - step2: Swap the sides: \(x^{2}-8x=0\) - step3: Factor the expression: \(x\left(x-8\right)=0\) - step4: Separate into possible cases: \(\begin{align}&x-8=0\\&x=0\end{align}\) - step5: Solve the equation: \(\begin{align}&x=8\\&x=0\end{align}\) - step6: Calculate: \(x_{1}=0,x_{2}=8\) Analyze the x intercept of the function \( y=x^{2}+2x-15 \) Function by following steps: - step0: Find the \(x\)-intercept/zero: \(y=x^{2}+2x-15\) - step1: Set \(y\)=0\(:\) \(0=x^{2}+2x-15\) - step2: Swap the sides: \(x^{2}+2x-15=0\) - step3: Factor the expression: \(\left(x-3\right)\left(x+5\right)=0\) - step4: Separate into possible cases: \(\begin{align}&x-3=0\\&x+5=0\end{align}\) - step5: Solve the equation: \(\begin{align}&x=3\\&x=-5\end{align}\) - step6: Calculate: \(x_{1}=-5,x_{2}=3\) Let's analyze each quadratic relation step by step. ### a) For the equation \( y = x^2 - 8x \) 1. **Axis of Symmetry**: The axis of symmetry for a quadratic equation in the form \( y = ax^2 + bx + c \) is given by the formula: \[ x = -\frac{b}{2a} \] Here, \( a = 1 \) and \( b = -8 \). \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \] So, the axis of symmetry is \( x = 4 \). 2. **Vertex**: The vertex can be found using the x-coordinate of the axis of symmetry. We substitute \( x = 4 \) back into the equation to find the y-coordinate: \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \] Therefore, the vertex is \( (4, -16) \). 3. **Y-Intercept**: The y-intercept occurs when \( x = 0 \): \[ y = 0^2 - 8(0) = 0 \] Thus, the y-intercept is \( (0, 0) \). 4. **Zeros**: The zeros (or x-intercepts) are found by setting \( y = 0 \): \[ 0 = x^2 - 8x \] Factoring gives: \[ x(x - 8) = 0 \] Thus, \( x_1 = 0 \) and \( x_2 = 8 \). ### Summary for \( y = x^2 - 8x \): - **Axis of Symmetry**: \( x = 4 \) - **Vertex**: \( (4, -16) \) - **Y-Intercept**: \( (0, 0) \) - **Zeros**: \( x_1 = 0, x_2 = 8 \) --- ### b) For the equation \( y = x^2 + 2x - 15 \) 1. **Axis of Symmetry**: Using the same formula: \[ x = -\frac{b}{2a} = -\frac{2}{2 \cdot 1} = -1 \] So, the axis of symmetry is \( x = -1 \). 2. **Vertex**: Substitute \( x = -1 \) back into the equation: \[ y = (-1)^2 + 2(-1) - 15 = 1 - 2 - 15 = -16 \] Therefore, the vertex is \( (-1, -16) \). 3. **Y-Intercept**: The y-intercept occurs when \( x = 0 \): \[ y = 0^2 + 2(0) - 15 = -15 \] Thus, the y-intercept is \( (0, -15) \). 4. **Zeros**: Set \( y = 0 \): \[ 0 = x^2 + 2x - 15 \] Factoring gives: \[ (x + 5)(x - 3) = 0 \] Thus, \( x_1 = -5 \) and \( x_2 = 3 \). ### Summary for \( y = x^2 + 2x - 15 \): - **Axis of Symmetry**: \( x = -1 \) - **Vertex**: \( (-1, -16) \) - **Y-Intercept**: \( (0, -15) \) - **Zeros**: \( x_1 = -5, x_2 = 3 \) ### Final Results: - **For \( y = x^2 - 8x \)**: - Axis of Symmetry: \( x = 4 \) - Vertex: \( (4, -16) \) - Y-Intercept: \( (0, 0) \) - Zeros: \( x_1 = 0, x_2 = 8 \) - **For \( y = x^2 + 2x - 15 \)**: - Axis of Symmetry: \( x = -1 \) - Vertex: \( (-1, -16) \) - Y-Intercept: \( (0, -15) \) - Zeros: \( x_1 = -5, x_2 = 3 \)