Answer :
To find the [tex]\( x \)[/tex]-intercepts of a parabola with vertex [tex]\((1, 1)\)[/tex] and [tex]\( y \)[/tex]-intercept [tex]\((0, -3)\)[/tex], we'll follow these steps:
1. Identify the vertex form of the parabola:
The vertex form of a parabola is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\( (h, k) \)[/tex] is the vertex. In this case, the vertex [tex]\((1, 1)\)[/tex] gives us [tex]\( h = 1 \)[/tex] and [tex]\( k = 1 \)[/tex]. So, the equation becomes:
[tex]\[ y = a(x - 1)^2 + 1 \][/tex]
2. Use the [tex]\( y \)[/tex]-intercept to find [tex]\( a \)[/tex]:
The [tex]\( y \)[/tex]-intercept is [tex]\((0, -3)\)[/tex]. This means when [tex]\( x = 0 \)[/tex], then [tex]\( y = -3 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = -3 \)[/tex] into the equation to solve for [tex]\( a \)[/tex]:
[tex]\[ -3 = a(0 - 1)^2 + 1 \][/tex]
Simplify the equation:
[tex]\[ -3 = a(1)^2 + 1 \][/tex]
[tex]\[ -3 = a + 1 \][/tex]
[tex]\[ -3 - 1 = a \][/tex]
[tex]\[ a = -4 \][/tex]
Now, the equation of the parabola is:
[tex]\[ y = -4(x - 1)^2 + 1 \][/tex]
3. Find the [tex]\( x \)[/tex]-intercepts:
The [tex]\( x \)[/tex]-intercepts occur where [tex]\( y = 0 \)[/tex]. Set [tex]\( y \)[/tex] to 0 in the equation:
[tex]\[ 0 = -4(x - 1)^2 + 1 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ -4(x - 1)^2 + 1 = 0 \][/tex]
[tex]\[ -4(x - 1)^2 = -1 \][/tex]
Divide by [tex]\(-4\)[/tex]:
[tex]\[ (x - 1)^2 = \frac{1}{4} \][/tex]
Take the square root of both sides:
[tex]\[ x - 1 = \pm \frac{1}{2} \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 1 + \frac{1}{2} \quad \text{or} \quad x = 1 - \frac{1}{2} \][/tex]
[tex]\[ x = 1.5 \quad \text{or} \quad x = 0.5 \][/tex]
4. Write the solutions:
The [tex]\( x \)[/tex]-intercepts are the points where [tex]\( y = 0 \)[/tex]:
[tex]\[ (0.5, 0) \quad \text{and} \quad (1.5, 0) \][/tex]
Therefore, the [tex]\( x \)[/tex]-intercepts of the parabola are:
[tex]\[ (0.5, 0), (1.5, 0) \][/tex]
1. Identify the vertex form of the parabola:
The vertex form of a parabola is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\( (h, k) \)[/tex] is the vertex. In this case, the vertex [tex]\((1, 1)\)[/tex] gives us [tex]\( h = 1 \)[/tex] and [tex]\( k = 1 \)[/tex]. So, the equation becomes:
[tex]\[ y = a(x - 1)^2 + 1 \][/tex]
2. Use the [tex]\( y \)[/tex]-intercept to find [tex]\( a \)[/tex]:
The [tex]\( y \)[/tex]-intercept is [tex]\((0, -3)\)[/tex]. This means when [tex]\( x = 0 \)[/tex], then [tex]\( y = -3 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = -3 \)[/tex] into the equation to solve for [tex]\( a \)[/tex]:
[tex]\[ -3 = a(0 - 1)^2 + 1 \][/tex]
Simplify the equation:
[tex]\[ -3 = a(1)^2 + 1 \][/tex]
[tex]\[ -3 = a + 1 \][/tex]
[tex]\[ -3 - 1 = a \][/tex]
[tex]\[ a = -4 \][/tex]
Now, the equation of the parabola is:
[tex]\[ y = -4(x - 1)^2 + 1 \][/tex]
3. Find the [tex]\( x \)[/tex]-intercepts:
The [tex]\( x \)[/tex]-intercepts occur where [tex]\( y = 0 \)[/tex]. Set [tex]\( y \)[/tex] to 0 in the equation:
[tex]\[ 0 = -4(x - 1)^2 + 1 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ -4(x - 1)^2 + 1 = 0 \][/tex]
[tex]\[ -4(x - 1)^2 = -1 \][/tex]
Divide by [tex]\(-4\)[/tex]:
[tex]\[ (x - 1)^2 = \frac{1}{4} \][/tex]
Take the square root of both sides:
[tex]\[ x - 1 = \pm \frac{1}{2} \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 1 + \frac{1}{2} \quad \text{or} \quad x = 1 - \frac{1}{2} \][/tex]
[tex]\[ x = 1.5 \quad \text{or} \quad x = 0.5 \][/tex]
4. Write the solutions:
The [tex]\( x \)[/tex]-intercepts are the points where [tex]\( y = 0 \)[/tex]:
[tex]\[ (0.5, 0) \quad \text{and} \quad (1.5, 0) \][/tex]
Therefore, the [tex]\( x \)[/tex]-intercepts of the parabola are:
[tex]\[ (0.5, 0), (1.5, 0) \][/tex]
