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Sketch a Quadratic Function with Given Vertex, Leading Coefficient and X-intercept Property
Mathematics
Grade 10 of Junior High School
Question Content
Sketch a quadratic function with the following properties: 1. Opens up (positive leading coefficient 9); 2. Vertex and x-intercept at (2, 0); 3. Touches the x-axis once
Correct Answer
The quadratic function is $y=9(x-2)^2$, and its graph is a parabola opening upward with vertex at (2, 0), tangent to the x-axis at that point.
Detailed Solution Steps
1
Step 1: Recall the vertex form of a quadratic function, which is $y=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola, and $a$ is the leading coefficient that determines the direction and width of the parabola.
2
Step 2: Substitute the given vertex $(2,0)$ into the vertex form: $h=2$, $k=0$, so the function becomes $y=a(x-2)^2$.
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Step 3: Use the given positive leading coefficient $a=9$ to complete the function: $y=9(x-2)^2$.
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Step 4: Verify the properties: Since $a=9>0$, the parabola opens upward; the vertex is $(2,0)$, and because the quadratic has a repeated root at $x=2$, the graph touches the x-axis exactly once at the vertex point.
Knowledge Points Involved
1
Vertex Form of a Quadratic Function
The vertex form of a quadratic function is $y=a(x-h)^2+k$, where $(h,k)$ represents the vertex of the parabola. This form is most useful when the vertex coordinates are known, as it directly incorporates them into the equation. $a$ controls the direction (positive $a$ opens upward, negative $a$ opens downward) and the width of the parabola.
2
X-intercepts of Quadratic Functions
X-intercepts of a quadratic function are the points where the graph crosses or touches the x-axis, corresponding to the roots of the quadratic equation $ax^2+bx+c=0$. A quadratic function touches the x-axis exactly once when it has a repeated (double) root, which occurs when the discriminant $b^2-4ac=0$, and the vertex lies on the x-axis.
3
Direction of Opening of a Parabola
The leading coefficient $a$ of a quadratic function determines the direction the parabola opens. If $a>0$, the parabola opens upward (U-shaped), and the vertex represents the minimum point of the function. If $a<0$, the parabola opens downward (inverted U-shaped), and the vertex represents the maximum point.
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