7. Determine the $$ y $$-intercept, zeros, equation of the axis of symmetry, and vertex of each quadratic relation. Then sketch its graph. a) $$ y=(x-5)(x+5) $$ b) $$ y=-(x-6)(x-2) $$ c) $$ y=2(x-1)(x+3) $$ d) $$ y=-0.5(x+4)^{2} $$

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**a) $$ y = (x-5)(x+5) $$**  
- **$$ y $$-intercept:**  
  Set $$ x=0 $$:  
  $$
  y = (0-5)(0+5) = (-5)(5) = -25.
  $$  
  So, the $$ y $$-intercept is $$ (0,-25) $$.

- **Zeros:**  
  Set $$ y=0 $$:  
  $$
  (x-5)(x+5)=0 \quad \Rightarrow \quad x-5=0 \quad \text{or} \quad x+5=0.
  $$  
  This gives:  
  $$
  x = 5 \quad \text{or} \quad x = -5.
  $$  
  The zeros are $$ (-5,0) $$ and $$ (5,0) $$.

- **Axis of symmetry:**  
  The axis of symmetry is the vertical line halfway between the zeros:  
  $$
  x = \frac{-5+5}{2} = 0.
  $$

- **Vertex:**  
  Substitute $$ x=0 $$ (the axis of symmetry) into the equation:  
  $$
  y = (0-5)(0+5) = -25.
  $$  
  The vertex is $$ (0,-25) $$.

---

**b) $$ y = -(x-6)(x-2) $$**  
- **$$ y $$-intercept:**  
  Set $$ x=0 $$:  
  $$
  y = - (0-6)(0-2) = -(-6)(-2) = - (12) = -12.
  $$  
  So, the $$ y $$-intercept is $$ (0,-12) $$.

- **Zeros:**  
  Set $$ y=0 $$:  
  $$
  -(x-6)(x-2)=0 \quad \Rightarrow \quad (x-6)(x-2)=0.
  $$  
  This gives:  
  $$
  x-6=0 \quad \Rightarrow \quad x=6,
  $$  
  or  
  $$
  x-2=0 \quad \Rightarrow \quad x=2.
  $$  
  The zeros are $$ (2,0) $$ and $$ (6,0) $$.

- **Axis of symmetry:**  
  The axis is halfway between $$ x=2 $$ and $$ x=6 $$:  
  $$
  x = \frac{2+6}{2} = 4.
  $$

- **Vertex:**  
  Evaluate $$ y $$ at $$ x=4 $$:  
  $$
  y = - (4-6)(4-2) = - (-2)(2) = -(-4) = 4.
  $$  
  The vertex is $$ (4,4) $$.  
  Since the coefficient is negative, the parabola opens downward.

---

**c) $$ y = 2(x-1)(x+3) $$**  
- **$$ y $$-intercept:**  
  Set $$ x=0 $$:  
  $$
  y = 2(0-1)(0+3) = 2(-1)(3) = -6.
  $$  
  So, the $$ y $$-intercept is $$ (0,-6) $$.

- **Zeros:**  
  Set $$ y=0 $$:  
  $$
  2(x-1)(x+3)=0 \quad \Rightarrow \quad (x-1)(x+3)=0.
  $$  
  This gives:  
  $$
  x-1=0 \quad \Rightarrow \quad x=1,
  $$
  or  
  $$
  x+3=0 \quad \Rightarrow \quad x=-3.
  $$  
  The zeros are $$ (1,0) $$ and $$ (-3,0) $$.

- **Axis of symmetry:**  
  The axis is halfway between $$ x=1 $$ and $$ x=-3 $$:  
  $$
  x = \frac{1+(-3)}{2} = \frac{-2}{2} = -1.
  $$

- **Vertex:**  
  Substitute $$ x=-1 $$:  
  $$
  y = 2(-1-1)(-1+3) = 2(-2)(2) = 2(-4) = -8.
  $$  
  The vertex is $$ (-1,-8) $$.  
  Since the coefficient $$ 2 $$ is positive, the parabola opens upward.

---

**d) $$ y = -0.5(x+4)^2 $$**  
- **$$ y $$-intercept:**  
  Set $$ x=0 $$:  
  $$
  y = -0.5(0+4)^2 = -0.5(16) = -8.
  $$  
  Thus, the $$ y $$-intercept is $$ (0,-8) $$.

- **Zeros:**  
  Set $$ y=0 $$:  
  $$
  -0.5(x+4)^2=0 \quad \Rightarrow \quad (x+4)^2=0.
  $$  
  Hence,  
  $$
  x+4=0 \quad \Rightarrow \quad x=-4.
  $$  
  There is a single (repeated) zero at $$ (-4,0) $$.

- **Axis of symmetry:**  
  The completed square form indicates the axis is  
  $$
  x = -4.
  $$

- **Vertex:**  
  In vertex form, $$ y=-0.5(x+4)^2 $$ has vertex $$ (-4, 0) $$.  
  With a negative coefficient, the parabola opens downward.

---

Each graph is a parabola with the identified intercepts, vertex, and axis of symmetry.  
- For part (a), the parabola opens upward with vertex at $$ (0,-25) $$.  
- For part (b), it opens downward with vertex at $$ (4,4) $$.  
- For part (c), it opens upward with vertex at $$ (-1,-8) $$.  
- For part (d), it opens downward with vertex at $$ (-4,0) $$.

These characteristics allow a sketch of each quadratic relation with correct positioning of intercepts and vertex.
**a) $$ y = (x-5)(x+5) $$** - **$$ y $$-intercept:** $$ (0, -25) $$ - **Zeros:** $$ (-5,0) $$ and $$ (5,0) $$ - **Axis of symmetry:** $$ x = 0 $$ - **Vertex:** $$ (0, -25) $$ **b) $$ y = -(x-6)(x-2) $$** - **$$ y $$-intercept:** $$ (0, -12) $$ - **Zeros:** $$ (2,0) $$ and $$ (6,0) $$ - **Axis of symmetry:** $$ x = 4 $$ - **Vertex:** $$ (4, 4) $$ **c) $$ y = 2(x-1)(x+3) $$** - **$$ y $$-intercept:** $$ (0, -6) $$ - **Zeros:** $$ (1,0) $$ and $$ (-3,0) $$ - **Axis of symmetry:** $$ x = -1 $$ - **Vertex:** $$ (-1, -8) $$ **d) $$ y = -0.5(x+4)^2 $$** - **$$ y $$-intercept:** $$ (0, -8) $$ - **Zero:** $$ (-4,0) $$ - **Axis of symmetry:** $$ x = -4 $$ - **Vertex:** $$ (-4, 0) $$ Each graph is a parabola with the specified intercepts, vertex, and axis of symmetry.

Determine the -intercept, zeros, equation of the
axis of symmetry, and vertex of each quadratic
relation. Then sketch its graph.
a)
b)
c)
d)

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Determine the -intercept, zeros, equation of the axis of symmetry, and vertex of each quadratic relation. Then sketch its graph. a) b) c) d)